math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 98.0%

Time: 6.9s

Alternatives: 7

Speedup: 1.4×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ t_1 := t_0 + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{if}\;x.im \leq -1.66 \cdot 10^{+162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -1.28 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 4.6 \cdot 10^{-100}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* (- x.re x.im) (+ x.re x.im))))
        (t_1 (+ t_0 (* x.re (* (* x.re x.im) 2.0)))))
   (if (<= x.im -1.66e+162)
     t_0
     (if (<= x.im -1.28e-76)
       t_1
       (if (<= x.im 4.6e-100)
         (* 3.0 (* x.re (* x.re x.im)))
         (if (<= x.im 5e+40) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	double t_1 = t_0 + (x_46_re * ((x_46_re * x_46_im) * 2.0));
	double tmp;
	if (x_46_im <= -1.66e+162) {
		tmp = t_0;
	} else if (x_46_im <= -1.28e-76) {
		tmp = t_1;
	} else if (x_46_im <= 4.6e-100) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 5e+40) {
		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_46im * ((x_46re - x_46im) * (x_46re + x_46im))
    t_1 = t_0 + (x_46re * ((x_46re * x_46im) * 2.0d0))
    if (x_46im <= (-1.66d+162)) then
        tmp = t_0
    else if (x_46im <= (-1.28d-76)) then
        tmp = t_1
    else if (x_46im <= 4.6d-100) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46im <= 5d+40) 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_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	double t_1 = t_0 + (x_46_re * ((x_46_re * x_46_im) * 2.0));
	double tmp;
	if (x_46_im <= -1.66e+162) {
		tmp = t_0;
	} else if (x_46_im <= -1.28e-76) {
		tmp = t_1;
	} else if (x_46_im <= 4.6e-100) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 5e+40) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))
	t_1 = t_0 + (x_46_re * ((x_46_re * x_46_im) * 2.0))
	tmp = 0
	if x_46_im <= -1.66e+162:
		tmp = t_0
	elif x_46_im <= -1.28e-76:
		tmp = t_1
	elif x_46_im <= 4.6e-100:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 5e+40:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im)))
	t_1 = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 2.0)))
	tmp = 0.0
	if (x_46_im <= -1.66e+162)
		tmp = t_0;
	elseif (x_46_im <= -1.28e-76)
		tmp = t_1;
	elseif (x_46_im <= 4.6e-100)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 5e+40)
		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_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	t_1 = t_0 + (x_46_re * ((x_46_re * x_46_im) * 2.0));
	tmp = 0.0;
	if (x_46_im <= -1.66e+162)
		tmp = t_0;
	elseif (x_46_im <= -1.28e-76)
		tmp = t_1;
	elseif (x_46_im <= 4.6e-100)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 5e+40)
		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[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.66e+162], t$95$0, If[LessEqual[x$46$im, -1.28e-76], t$95$1, If[LessEqual[x$46$im, 4.6e-100], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5e+40], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.66000000000000003e162 or 5.00000000000000003e40 < x.im

   1. Initial program 69.4%

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

      1. expm1-log1p-u 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      2. expm1-udef 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      3. *-commutative 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      4. *-commutative 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      5. count-2 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 57.6%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)} - 1\right) \]

      11. flip-+ 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      13. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   3. Applied egg-rr 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   5. Simplified 85.9%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.66000000000000003e162 < x.im < -1.28e-76 or 4.59999999999999989e-100 < x.im < 5.00000000000000003e40

   1. Initial program 97.1%

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

      1. difference-of-squares 83.4%

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

      2. *-commutative 83.4%

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

   3. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. *-commutative 99.7%

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

      6. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   if -1.28e-76 < x.im < 4.59999999999999989e-100

   1. Initial program 83.2%

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

   2. Taylor expanded in x.re around inf 83.1%

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

   4. Simplified 83.2%

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

      1. add-sqr-sqrt 66.7%

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

      2. pow2 66.7%

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

      3. *-commutative 66.7%

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

      4. sqrt-prod 40.3%

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

      5. unpow2 40.3%

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

      6. sqrt-prod 26.1%

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

      7. add-sqr-sqrt 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. unpow2 48.2%

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

      2. *-commutative 48.2%

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

      3. associate-*r* 48.2%

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

      4. associate-*r* 48.2%

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

      5. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.66 \cdot 10^{+162}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -1.28 \cdot 10^{-76}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \leq 4.6 \cdot 10^{-100}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]

Alternative 2: 97.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.9%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x.re around 0 98.6%

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

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

   1. Initial program 0.0%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      3. *-commutative 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      4. *-commutative 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      5. count-2 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)} - 1\right) \]

      11. flip-+ 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      13. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   3. Applied egg-rr 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   5. Simplified 50.0%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

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

Alternative 3: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{-76} \lor \neg \left(x.im \leq 6.5 \cdot 10^{-82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -4.2e-76) (not (<= x.im 6.5e-82)))
   (* x.im (* (- x.re x.im) (+ x.re x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4.2e-76) || !(x_46_im <= 6.5e-82)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	} else {
		tmp = 3.0 * (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) :: tmp
    if ((x_46im <= (-4.2d-76)) .or. (.not. (x_46im <= 6.5d-82))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46re + x_46im))
    else
        tmp = 3.0d0 * (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 tmp;
	if ((x_46_im <= -4.2e-76) || !(x_46_im <= 6.5e-82)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -4.2e-76) or not (x_46_im <= 6.5e-82):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -4.2e-76) || !(x_46_im <= 6.5e-82))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im)));
	else
		tmp = Float64(3.0 * 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)
	tmp = 0.0;
	if ((x_46_im <= -4.2e-76) || ~((x_46_im <= 6.5e-82)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -4.2e-76], N[Not[LessEqual[x$46$im, 6.5e-82]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.19999999999999985e-76 or 6.4999999999999997e-82 < x.im

   1. Initial program 81.9%

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

      1. expm1-log1p-u 60.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

      2. expm1-udef 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

      3. *-commutative 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

      4. *-commutative 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

      5. count-2 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      6. *-commutative 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      7. associate-*r* 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

      8. associate-*r* 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

      9. *-commutative 59.5%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

      10. count-2 59.5%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)} - 1\right) \]

      11. flip-+ 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      13. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   3. Applied egg-rr 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

   5. Simplified 85.6%

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

      1. difference-of-squares 94.5%

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

      2. *-commutative 94.5%

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

   7. Applied egg-rr 94.5%

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

   if -4.19999999999999985e-76 < x.im < 6.4999999999999997e-82

   1. Initial program 84.0%

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

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

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

   4. Simplified 84.0%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

      3. *-commutative 68.3%

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

      4. sqrt-prod 43.2%

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

      5. unpow2 43.2%

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

      6. sqrt-prod 27.8%

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

      7. add-sqr-sqrt 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow2 50.7%

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

      2. *-commutative 50.7%

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

      3. associate-*r* 50.7%

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

      4. associate-*r* 50.7%

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

      5. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{-76} \lor \neg \left(x.im \leq 6.5 \cdot 10^{-82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 4: 56.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 3.0 * (x_46_re * (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 = 3.0d0 * (x_46re * (x_46re * x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

2. Taylor expanded in x.re around inf 49.7%

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

4. Simplified 49.8%

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

   1. add-sqr-sqrt 35.7%

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

   2. pow2 35.7%

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

   3. *-commutative 35.7%

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

   4. sqrt-prod 25.8%

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

   5. unpow2 25.8%

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

   6. sqrt-prod 14.5%

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

   7. add-sqr-sqrt 28.7%

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

6. Applied egg-rr 28.7%

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

   1. unpow2 28.7%

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

   2. *-commutative 28.7%

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

   3. associate-*r* 28.7%

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

   4. associate-*r* 28.7%

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

   5. add-sqr-sqrt 55.9%

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

8. Applied egg-rr 55.9%

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

9. Final simplification 55.9%

   \[\leadsto 3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \]

Alternative 5: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

2. Taylor expanded in x.re around inf 49.7%

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

3. Taylor expanded in x.im around 0 49.8%

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

5. Simplified 2.6%

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

6. Final simplification 2.6%

   \[\leadsto -3 \]

Alternative 6: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -0.3333333333333333)

C

double code(double x_46_re, double x_46_im) {
	return -0.3333333333333333;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -0.3333333333333333d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -0.3333333333333333;
}

Python

def code(x_46_re, x_46_im):
	return -0.3333333333333333

Julia

function code(x_46_re, x_46_im)
	return -0.3333333333333333
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -0.3333333333333333;
end

Wolfram

code[x$46$re_, x$46$im_] := -0.3333333333333333

Derivation

1. Initial program 82.7%

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

2. Taylor expanded in x.re around inf 49.7%

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

4. Simplified 49.8%

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

   1. associate-*r* 49.7%

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

   2. metadata-eval 49.7%

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

   3. distribute-rgt1-in 49.7%

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

   4. *-commutative 49.7%

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

   5. flip-+ 38.6%

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

   6. associate-*r/ 36.3%

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

   7. difference-of-squares 39.4%

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

   8. *-un-lft-identity 39.4%

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

   9. distribute-rgt-out 39.4%

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

   10. metadata-eval 39.4%

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

   11. *-un-lft-identity 39.4%

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

   12. distribute-rgt-out-- 39.4%

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

   13. metadata-eval 39.4%

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

   14. *-un-lft-identity 39.4%

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

   15. distribute-rgt-out-- 39.4%

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

   16. metadata-eval 39.4%

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

6. Applied egg-rr 39.4%

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

8. Simplified 2.6%

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

10. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{-0.3333333333333333} \]

11. Final simplification 2.6%

    \[\leadsto -0.3333333333333333 \]

Alternative 7: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 0.3333333333333333)

C

double code(double x_46_re, double x_46_im) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.3333333333333333d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.3333333333333333;
}

Python

def code(x_46_re, x_46_im):
	return 0.3333333333333333

Julia

function code(x_46_re, x_46_im)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.3333333333333333;
end

Wolfram

code[x$46$re_, x$46$im_] := 0.3333333333333333

Derivation

1. Initial program 82.7%

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

2. Taylor expanded in x.re around inf 49.7%

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

4. Simplified 49.8%

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

   1. associate-*r* 49.7%

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

   2. metadata-eval 49.7%

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

   3. distribute-rgt1-in 49.7%

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

   4. *-commutative 49.7%

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

   5. flip-+ 38.6%

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

   6. associate-*r/ 36.3%

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

   7. difference-of-squares 39.4%

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

   8. *-un-lft-identity 39.4%

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

   9. distribute-rgt-out 39.4%

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

   10. metadata-eval 39.4%

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

   11. *-un-lft-identity 39.4%

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

   12. distribute-rgt-out-- 39.4%

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

   13. metadata-eval 39.4%

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

   14. *-un-lft-identity 39.4%

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

   15. distribute-rgt-out-- 39.4%

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

   16. metadata-eval 39.4%

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

6. Applied egg-rr 39.4%

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

8. Simplified 2.6%

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

10. Applied egg-rr 2.8%

    \[\leadsto \color{blue}{0.3333333333333333} \]

11. Final simplification 2.8%

    \[\leadsto 0.3333333333333333 \]

Developer target: 91.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023333 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

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

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