math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.8%

Time: 8.6s

Alternatives: 11

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: 82.7% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 9 \cdot 10^{+96}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.im\_m - x.im\_m\right) + x.re\_m \cdot \left(x.im\_m + x.im\_m \cdot 2\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 9e+96)
    (-
     (*
      x.re_m
      (+ (* x.im_m (- x.im_m x.im_m)) (* x.re_m (+ x.im_m (* x.im_m 2.0)))))
     (pow x.im_m 3.0))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9e+96) {
		tmp = (x_46_re_m * ((x_46_im_m * (x_46_im_m - x_46_im_m)) + (x_46_re_m * (x_46_im_m + (x_46_im_m * 2.0))))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 9d+96) then
        tmp = (x_46re_m * ((x_46im_m * (x_46im_m - x_46im_m)) + (x_46re_m * (x_46im_m + (x_46im_m * 2.0d0))))) - (x_46im_m ** 3.0d0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9e+96) {
		tmp = (x_46_re_m * ((x_46_im_m * (x_46_im_m - x_46_im_m)) + (x_46_re_m * (x_46_im_m + (x_46_im_m * 2.0))))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 9e+96:
		tmp = (x_46_re_m * ((x_46_im_m * (x_46_im_m - x_46_im_m)) + (x_46_re_m * (x_46_im_m + (x_46_im_m * 2.0))))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 9e+96)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * Float64(x_46_im_m - x_46_im_m)) + Float64(x_46_re_m * Float64(x_46_im_m + Float64(x_46_im_m * 2.0))))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 9e+96)
		tmp = (x_46_re_m * ((x_46_im_m * (x_46_im_m - x_46_im_m)) + (x_46_re_m * (x_46_im_m + (x_46_im_m * 2.0))))) - (x_46_im_m ^ 3.0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 9e+96], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * N[(x$46$im$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im$95$m + N[(x$46$im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.99999999999999914e96

   1. Initial program 84.5%

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

      1. difference-of-squares 87.4%

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

      2. *-commutative 87.4%

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

   4. Applied egg-rr 87.4%

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

   5. Taylor expanded in x.re around 0 87.4%

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

   if 8.99999999999999914e96 < x.im

   1. Initial program 55.3%

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

      1. difference-of-squares 61.7%

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

      2. *-commutative 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in x.re around 0 55.3%

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

      1. add-log-exp 51.5%

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

      2. +-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. exp-sum 42.8%

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

      5. *-commutative 42.8%

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

      6. exp-prod 47.0%

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

      7. *-commutative 47.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 51.5%

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

      10. sqr-neg 51.5%

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

      11. sqrt-unprod 43.0%

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

      12. add-sqr-sqrt 49.4%

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

      13. sub-neg 49.4%

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

      14. +-inverses 89.8%

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

      15. metadata-eval 89.8%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im - x.im\right) + x.re \cdot \left(x.im + x.im \cdot 2\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+96}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+96)
    (- (* 3.0 (* x.re_m (* x.im_m x.re_m))) (pow x.im_m 3.0))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+96) {
		tmp = (3.0 * (x_46_re_m * (x_46_im_m * x_46_re_m))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+96) then
        tmp = (3.0d0 * (x_46re_m * (x_46im_m * x_46re_m))) - (x_46im_m ** 3.0d0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+96) {
		tmp = (3.0 * (x_46_re_m * (x_46_im_m * x_46_re_m))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+96:
		tmp = (3.0 * (x_46_re_m * (x_46_im_m * x_46_re_m))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+96)
		tmp = Float64(Float64(3.0 * Float64(x_46_re_m * Float64(x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+96)
		tmp = (3.0 * (x_46_re_m * (x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+96], N[(N[(3.0 * N[(x$46$re$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000004e96

   1. Initial program 84.5%

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

   3. Simplified 87.4%

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

   if 5.0000000000000004e96 < x.im

   1. Initial program 55.3%

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

      1. difference-of-squares 61.7%

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

      2. *-commutative 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in x.re around 0 55.3%

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

      1. add-log-exp 51.5%

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

      2. +-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. exp-sum 42.8%

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

      5. *-commutative 42.8%

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

      6. exp-prod 47.0%

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

      7. *-commutative 47.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 51.5%

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

      10. sqr-neg 51.5%

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

      11. sqrt-unprod 43.0%

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

      12. add-sqr-sqrt 49.4%

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

      13. sub-neg 49.4%

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

      14. +-inverses 89.8%

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

      15. metadata-eval 89.8%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+96}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+96}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(3 \cdot x.re\_m\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+96)
    (- (* x.re_m (* x.im_m (* 3.0 x.re_m))) (pow x.im_m 3.0))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+96) {
		tmp = (x_46_re_m * (x_46_im_m * (3.0 * x_46_re_m))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+96) then
        tmp = (x_46re_m * (x_46im_m * (3.0d0 * x_46re_m))) - (x_46im_m ** 3.0d0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+96) {
		tmp = (x_46_re_m * (x_46_im_m * (3.0 * x_46_re_m))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+96:
		tmp = (x_46_re_m * (x_46_im_m * (3.0 * x_46_re_m))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+96)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_im_m * Float64(3.0 * x_46_re_m))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+96)
		tmp = (x_46_re_m * (x_46_im_m * (3.0 * x_46_re_m))) - (x_46_im_m ^ 3.0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+96], N[(N[(x$46$re$95$m * N[(x$46$im$95$m * N[(3.0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000004e96

   1. Initial program 84.5%

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

   3. Simplified 87.4%

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

   if 5.0000000000000004e96 < x.im

   1. Initial program 55.3%

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

      1. difference-of-squares 61.7%

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

      2. *-commutative 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in x.re around 0 55.3%

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

      1. add-log-exp 51.5%

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

      2. +-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. exp-sum 42.8%

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

      5. *-commutative 42.8%

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

      6. exp-prod 47.0%

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

      7. *-commutative 47.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 51.5%

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

      10. sqr-neg 51.5%

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

      11. sqrt-unprod 43.0%

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

      12. add-sqr-sqrt 49.4%

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

      13. sub-neg 49.4%

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

      14. +-inverses 89.8%

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

      15. metadata-eval 89.8%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 8 \cdot 10^{+96}:\\ \;\;\;\;x.re\_m \cdot \left(3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 8e+96)
    (- (* x.re_m (* 3.0 (* x.im_m x.re_m))) (pow x.im_m 3.0))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 8e+96) {
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 8d+96) then
        tmp = (x_46re_m * (3.0d0 * (x_46im_m * x_46re_m))) - (x_46im_m ** 3.0d0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 8e+96) {
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 8e+96:
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 8e+96)
		tmp = Float64(Float64(x_46_re_m * Float64(3.0 * Float64(x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 8e+96)
		tmp = (x_46_re_m * (3.0 * (x_46_im_m * x_46_re_m))) - (x_46_im_m ^ 3.0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 8e+96], N[(N[(x$46$re$95$m * N[(3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.0000000000000004e96

   1. Initial program 84.5%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in x.im around 0 87.4%

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

   if 8.0000000000000004e96 < x.im

   1. Initial program 55.3%

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

      1. difference-of-squares 61.7%

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

      2. *-commutative 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in x.re around 0 55.3%

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

      1. add-log-exp 51.5%

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

      2. +-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. exp-sum 42.8%

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

      5. *-commutative 42.8%

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

      6. exp-prod 47.0%

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

      7. *-commutative 47.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 51.5%

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

      10. sqr-neg 51.5%

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

      11. sqrt-unprod 43.0%

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

      12. add-sqr-sqrt 49.4%

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

      13. sub-neg 49.4%

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

      14. +-inverses 89.8%

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

      15. metadata-eval 89.8%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 8 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 + x.re\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \leq 10^{+298}:\\ \;\;\;\;x.re\_m \cdot \left(2 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot \left(x.im\_m + x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))))
   (*
    x.im_s
    (if (<= (+ t_0 (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m)))) 1e+298)
      (+ (* x.re_m (* 2.0 (* x.im_m x.re_m))) t_0)
      (* (* x.im_m (- x.re_m x.im_m)) (+ x.im_m x.re_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= 1e+298) {
		tmp = (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m))) + t_0;
	} else {
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))
    if ((t_0 + (x_46re_m * ((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)))) <= 1d+298) then
        tmp = (x_46re_m * (2.0d0 * (x_46im_m * x_46re_m))) + t_0
    else
        tmp = (x_46im_m * (x_46re_m - x_46im_m)) * (x_46im_m + x_46re_m)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= 1e+298) {
		tmp = (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m))) + t_0;
	} else {
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))
	tmp = 0
	if (t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= 1e+298:
		tmp = (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m))) + t_0
	else:
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)))
	tmp = 0.0
	if (Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)))) <= 1e+298)
		tmp = Float64(Float64(x_46_re_m * Float64(2.0 * Float64(x_46_im_m * x_46_re_m))) + t_0);
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)) * Float64(x_46_im_m + x_46_re_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	tmp = 0.0;
	if ((t_0 + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)))) <= 1e+298)
		tmp = (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m))) + t_0;
	else
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+298], N[(N[(x$46$re$95$m * N[(2.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $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)) < 9.9999999999999996e297

   1. Initial program 94.6%

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

      1. *-commutative 95.3%

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

      2. count-2 95.3%

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

      3. *-commutative 95.3%

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

   4. Applied egg-rr 94.6%

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

   if 9.9999999999999996e297 < (+.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 41.9%

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

      1. difference-of-squares 53.9%

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

      2. *-commutative 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. *-commutative 53.9%

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

      2. distribute-rgt-in 47.2%

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

      3. distribute-rgt-in 39.2%

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

   6. Applied egg-rr 39.2%

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

   8. Applied egg-rr 93.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+298}:\\ \;\;\;\;x.re \cdot \left(2 \cdot \left(x.im \cdot x.re\right)\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10^{+96}:\\ \;\;\;\;\left(t\_0 + \left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\right) + x.re\_m \cdot \left(2 \cdot \left(x.im\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (* x.im_m (- x.re_m x.im_m)))))
   (*
    x.im_s
    (if (<= x.im_m 1e+96)
      (+
       (+ t_0 (* (- x.re_m x.im_m) (* x.im_m x.re_m)))
       (* x.re_m (* 2.0 (* x.im_m x.re_m))))
      t_0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	double tmp;
	if (x_46_im_m <= 1e+96) {
		tmp = (t_0 + ((x_46_re_m - x_46_im_m) * (x_46_im_m * x_46_re_m))) + (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m)));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    if (x_46im_m <= 1d+96) then
        tmp = (t_0 + ((x_46re_m - x_46im_m) * (x_46im_m * x_46re_m))) + (x_46re_m * (2.0d0 * (x_46im_m * x_46re_m)))
    else
        tmp = t_0
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	double tmp;
	if (x_46_im_m <= 1e+96) {
		tmp = (t_0 + ((x_46_re_m - x_46_im_m) * (x_46_im_m * x_46_re_m))) + (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m)));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	tmp = 0
	if x_46_im_m <= 1e+96:
		tmp = (t_0 + ((x_46_re_m - x_46_im_m) * (x_46_im_m * x_46_re_m))) + (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m)))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)))
	tmp = 0.0
	if (x_46_im_m <= 1e+96)
		tmp = Float64(Float64(t_0 + Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m * x_46_re_m))) + Float64(x_46_re_m * Float64(2.0 * Float64(x_46_im_m * x_46_re_m))));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	tmp = 0.0;
	if (x_46_im_m <= 1e+96)
		tmp = (t_0 + ((x_46_re_m - x_46_im_m) * (x_46_im_m * x_46_re_m))) + (x_46_re_m * (2.0 * (x_46_im_m * x_46_re_m)));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1e+96], N[(N[(t$95$0 + N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(2.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.00000000000000005e96

   1. Initial program 84.5%

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

      1. difference-of-squares 87.4%

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

      2. *-commutative 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. *-commutative 87.4%

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

      2. distribute-rgt-in 85.4%

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

      3. distribute-rgt-in 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x.re around 0 84.9%

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

      1. *-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unpow2 84.9%

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

      4. distribute-rgt-neg-in 84.9%

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

      5. distribute-lft-in 86.8%

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

      6. +-commutative 86.8%

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

      7. sub-neg 86.8%

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

      8. *-commutative 86.8%

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

      9. associate-*r* 87.8%

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

   9. Simplified 87.8%

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

       1. *-commutative 87.8%

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

       2. count-2 87.8%

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

       3. *-commutative 87.8%

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

   11. Applied egg-rr 87.8%

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

   if 1.00000000000000005e96 < x.im

   1. Initial program 55.3%

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

      1. difference-of-squares 61.7%

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

      2. *-commutative 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in x.re around 0 55.3%

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

      1. add-log-exp 51.5%

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

      2. +-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. exp-sum 42.8%

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

      5. *-commutative 42.8%

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

      6. exp-prod 47.0%

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

      7. *-commutative 47.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 51.5%

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

      10. sqr-neg 51.5%

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

      11. sqrt-unprod 43.0%

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

      12. add-sqr-sqrt 49.4%

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

      13. sub-neg 49.4%

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

      14. +-inverses 89.8%

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

      15. metadata-eval 89.8%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 10^{+96}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + \left(x.re - x.im\right) \cdot \left(x.im \cdot x.re\right)\right) + x.re \cdot \left(2 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot \left(x.im\_m + x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.5e-95)
    (* (* x.re_m x.re_m) (* x.im_m 3.0))
    (* (* x.im_m (- x.re_m x.im_m)) (+ x.im_m x.re_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.5e-95) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.5d-95) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else
        tmp = (x_46im_m * (x_46re_m - x_46im_m)) * (x_46im_m + x_46re_m)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.5e-95) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.5e-95:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	else:
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.5e-95)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)) * Float64(x_46_im_m + x_46_re_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.5e-95)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	else
		tmp = (x_46_im_m * (x_46_re_m - x_46_im_m)) * (x_46_im_m + x_46_re_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.5e-95], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5e-95

   1. Initial program 82.5%

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

   3. Taylor expanded in x.re around inf 62.4%

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

      1. unpow2 62.4%

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

   5. Applied egg-rr 62.4%

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

      1. distribute-rgt1-in 62.4%

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

      2. metadata-eval 62.4%

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

      3. *-commutative 62.4%

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

   7. Applied egg-rr 62.4%

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

   if 1.5e-95 < x.im

   1. Initial program 71.3%

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

      1. difference-of-squares 75.2%

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

      2. *-commutative 75.2%

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

   4. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

      2. distribute-rgt-in 71.3%

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

      3. distribute-rgt-in 68.7%

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

   6. Applied egg-rr 68.7%

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

   8. Applied egg-rr 92.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.6 \cdot 10^{-85}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2.6e-85)
    (* (* x.re_m x.re_m) (* x.im_m 3.0))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.6e-85) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2.6d-85) then
        tmp = (x_46re_m * x_46re_m) * (x_46im_m * 3.0d0)
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.6e-85) {
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2.6e-85:
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0)
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.6e-85)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2.6e-85)
		tmp = (x_46_re_m * x_46_re_m) * (x_46_im_m * 3.0);
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2.6e-85], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.60000000000000011e-85

   1. Initial program 82.6%

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

   3. Taylor expanded in x.re around inf 62.6%

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

      1. unpow2 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. distribute-rgt1-in 62.6%

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

      2. metadata-eval 62.6%

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

      3. *-commutative 62.6%

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

   7. Applied egg-rr 62.6%

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

   if 2.60000000000000011e-85 < x.im

   1. Initial program 71.0%

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

      1. difference-of-squares 74.9%

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

      2. *-commutative 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in x.re around 0 64.7%

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

      1. add-log-exp 40.2%

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

      2. +-commutative 40.2%

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

      3. *-commutative 40.2%

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

      4. exp-sum 30.4%

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

      5. *-commutative 30.4%

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

      6. exp-prod 33.1%

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

      7. *-commutative 33.1%

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

      8. add-sqr-sqrt 33.1%

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

      9. sqrt-unprod 36.0%

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

      10. sqr-neg 36.0%

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

      11. sqrt-unprod 27.7%

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

      12. add-sqr-sqrt 34.7%

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

      13. sub-neg 34.7%

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

      14. +-inverses 59.7%

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

      15. metadata-eval 59.7%

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

   7. Applied egg-rr 80.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.6 \cdot 10^{-85}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.35e-85)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (* x.im_m (* x.im_m (- x.re_m x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.35e-85) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.35d-85) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.35e-85) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.35e-85:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.35e-85)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.35e-85)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.35e-85], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.3500000000000001e-85

   1. Initial program 82.6%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in x.re around inf 62.6%

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

      1. unpow2 62.6%

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

   7. Applied egg-rr 62.6%

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

   if 1.3500000000000001e-85 < x.im

   1. Initial program 71.0%

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

      1. difference-of-squares 74.9%

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

      2. *-commutative 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in x.re around 0 64.7%

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

      1. add-log-exp 40.2%

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

      2. +-commutative 40.2%

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

      3. *-commutative 40.2%

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

      4. exp-sum 30.4%

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

      5. *-commutative 30.4%

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

      6. exp-prod 33.1%

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

      7. *-commutative 33.1%

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

      8. add-sqr-sqrt 33.1%

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

      9. sqrt-unprod 36.0%

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

      10. sqr-neg 36.0%

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

      11. sqrt-unprod 27.7%

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

      12. add-sqr-sqrt 34.7%

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

      13. sub-neg 34.7%

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

      14. +-inverses 59.7%

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

      15. metadata-eval 59.7%

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

   7. Applied egg-rr 80.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (* x.im_s (* 3.0 (* x.im_m (* x.re_m x.re_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (3.0d0 * (x_46im_m * (x_46re_m * x_46re_m)))
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)))

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m))))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Simplified 80.7%

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

5. Taylor expanded in x.re around inf 50.6%

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

   1. unpow2 50.6%

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

7. Applied egg-rr 50.6%

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

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot -3 \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s -3.0))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-3.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -3.0

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * -3.0), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in x.re around 0 57.4%

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

5. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]
6. Add Preprocessing

Developer Target 1: 91.9% 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 2024113 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 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)))