math.cube on complex, real part

Percentage Accurate: 82.7% → 99.8%

Time: 10.7s

Alternatives: 11

Speedup: 1.2×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+99}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) - x.im \cdot \left(x.re\_m + x.re\_m \cdot 2\right)\right) + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+99)
    (+
     (*
      x.im
      (- (* x.re_m (- x.re_m x.re_m)) (* x.im (+ x.re_m (* x.re_m 2.0)))))
     (pow x.re_m 3.0))
    (- (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) x.im))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+99) {
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + pow(x_46_re_m, 3.0);
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 5d+99) then
        tmp = (x_46im * ((x_46re_m * (x_46re_m - x_46re_m)) - (x_46im * (x_46re_m + (x_46re_m * 2.0d0))))) + (x_46re_m ** 3.0d0)
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - x_46im
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+99) {
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 5e+99:
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5e+99)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) - Float64(x_46_im * Float64(x_46_re_m + Float64(x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 5e+99)
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0);
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e+99], N[(N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m + N[(x$46$re$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.00000000000000008e99

   1. Initial program 85.1%

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

      1. difference-of-squares 86.1%

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

      2. *-commutative 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x.im around 0 91.5%

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

   if 5.00000000000000008e99 < x.re

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. distribute-lft-in 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. metadata-eval 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. pow1 75.0%

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

      2. *-commutative 75.0%

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

      3. add-log-exp 17.5%

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

      4. exp-lft-sqr 17.5%

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

      5. exp-sum 17.5%

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

      6. add-log-exp 75.0%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 90.0%

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

      1. difference-of-squares 82.5%

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

      2. *-commutative 82.5%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+99}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) - x.im \cdot \left(x.re + x.re \cdot 2\right)\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+99}:\\ \;\;\;\;{x.re\_m}^{3} + x.im \cdot \left(-3 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+99)
    (+ (pow x.re_m 3.0) (* x.im (* -3.0 (* x.re_m x.im))))
    (- (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) x.im))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+99) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_im * (-3.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 5d+99) then
        tmp = (x_46re_m ** 3.0d0) + (x_46im * ((-3.0d0) * (x_46re_m * x_46im)))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - x_46im
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+99) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_im * (-3.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 5e+99:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_im * (-3.0 * (x_46_re_m * x_46_im)))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5e+99)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_im * Float64(-3.0 * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 5e+99)
		tmp = (x_46_re_m ^ 3.0) + (x_46_im * (-3.0 * (x_46_re_m * x_46_im)));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e+99], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$im * N[(-3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.00000000000000008e99

   1. Initial program 85.1%

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

      1. difference-of-squares 86.1%

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

      2. *-commutative 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x.im around 0 91.5%

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

   6. Taylor expanded in x.re around 0 91.5%

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

   if 5.00000000000000008e99 < x.re

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. distribute-lft-in 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. metadata-eval 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. pow1 75.0%

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

      2. *-commutative 75.0%

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

      3. add-log-exp 17.5%

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

      4. exp-lft-sqr 17.5%

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

      5. exp-sum 17.5%

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

      6. add-log-exp 75.0%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 90.0%

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

      1. difference-of-squares 82.5%

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

      2. *-commutative 82.5%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+99}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ t_1 := t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3 - x.re\_m\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;t\_0 - x.im \cdot \left(2 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))))
        (t_1 (- t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_1 (- INFINITY))
      (* x.im (* x.re_m (- (* x.im -3.0) x.re_m)))
      (if (<= t_1 1e+299)
        (- t_0 (* x.im (* 2.0 (* x.re_m x.im))))
        (- (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	} else if (t_1 <= 1e+299) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	} else if (t_1 <= 1e+299) {
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m))
	elif t_1 <= 1e+299:
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	t_1 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(Float64(x_46_im * -3.0) - x_46_re_m)));
	elseif (t_1 <= 1e+299)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(2.0 * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	elseif (t_1 <= 1e+299)
		tmp = t_0 - (x_46_im * (2.0 * (x_46_re_m * x_46_im)));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(x$46$im * N[(x$46$re$95$m * N[(N[(x$46$im * -3.0), $MachinePrecision] - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(t$95$0 - N[(x$46$im * N[(2.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.2%

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

      1. difference-of-squares 84.2%

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

      2. *-commutative 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in x.re around -inf 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x.im around inf 45.6%

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

   9. Taylor expanded in x.re around 0 50.7%

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

       1. associate--l+ 50.7%

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

       2. distribute-rgt-out-- 50.7%

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

       3. unpow2 50.7%

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

       4. metadata-eval 50.7%

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

       5. associate-*r* 50.7%

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

       6. distribute-rgt-in 48.2%

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

       7. mul-1-neg 48.2%

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

       8. distribute-rgt-neg-in 48.2%

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

       9. associate-*r* 57.5%

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

       10. associate-*r* 73.2%

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

       11. *-commutative 73.2%

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

       12. associate-*r* 73.1%

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

       13. *-commutative 73.1%

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

       14. associate-*r* 73.2%

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

       15. metadata-eval 73.2%

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

       16. distribute-rgt-out-- 73.2%

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

       17. distribute-lft-in 78.1%

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

   11. Simplified 78.1%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.0000000000000001e299

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. distribute-rgt-out 99.8%

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

      5. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   if 1.0000000000000001e299 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

      2. *-un-lft-identity 52.3%

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

      3. distribute-lft-in 52.3%

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

      4. distribute-rgt-out 52.3%

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

      5. metadata-eval 52.3%

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

   4. Applied egg-rr 52.3%

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

      1. pow1 52.3%

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

      2. *-commutative 52.3%

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

      3. add-log-exp 21.2%

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

      4. exp-lft-sqr 21.2%

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

      5. exp-sum 21.2%

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

      6. add-log-exp 52.3%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 71.2%

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

      1. difference-of-squares 59.0%

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

      2. *-commutative 59.0%

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

   10. Applied egg-rr 86.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -\infty:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3 - x.re\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+299}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(2 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3 - x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.05e-54)
    (* x.im (* x.re_m (- (* x.im -3.0) x.re_m)))
    (- (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) x.im))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.05e-54) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1.05d-54) then
        tmp = x_46im * (x_46re_m * ((x_46im * (-3.0d0)) - x_46re_m))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - x_46im
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.05e-54) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1.05e-54:
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.05e-54)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(Float64(x_46_im * -3.0) - x_46_re_m)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1.05e-54)
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - x_46_im;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.05e-54], N[(x$46$im * N[(x$46$re$95$m * N[(N[(x$46$im * -3.0), $MachinePrecision] - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.05e-54

   1. Initial program 83.0%

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

      1. difference-of-squares 84.0%

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

      2. *-commutative 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x.re around -inf 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in x.im around inf 56.3%

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

   9. Taylor expanded in x.re around 0 61.0%

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

       1. associate--l+ 61.0%

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

       2. distribute-rgt-out-- 61.0%

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

       3. unpow2 61.0%

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

       4. metadata-eval 61.0%

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

       5. associate-*r* 60.9%

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

       6. distribute-rgt-in 60.4%

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

       7. mul-1-neg 60.4%

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

       8. distribute-rgt-neg-in 60.4%

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

       9. associate-*r* 62.4%

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

       10. associate-*r* 71.3%

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

       11. *-commutative 71.3%

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

       12. associate-*r* 71.3%

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

       13. *-commutative 71.3%

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

       14. associate-*r* 71.4%

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

       15. metadata-eval 71.4%

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

       16. distribute-rgt-out-- 71.4%

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

       17. distribute-lft-in 72.4%

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

   11. Simplified 74.0%

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

   if 1.05e-54 < x.re

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. distribute-lft-in 85.2%

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

      4. distribute-rgt-out 85.2%

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

      5. metadata-eval 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. pow1 85.2%

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

      2. *-commutative 85.2%

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

      3. add-log-exp 45.9%

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

      4. exp-lft-sqr 45.9%

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

      5. exp-sum 45.9%

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

      6. add-log-exp 85.2%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 86.4%

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

      1. difference-of-squares 89.6%

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

      2. *-commutative 89.6%

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

   10. Applied egg-rr 92.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3 - x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 19.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;-1 + \left(1 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 3.8e-20) (+ -1.0 (- 1.0 x.im)) (* x.re_m (* x.im -27.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.8e-20) {
		tmp = -1.0 + (1.0 - x_46_im);
	} else {
		tmp = x_46_re_m * (x_46_im * -27.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3.8d-20) then
        tmp = (-1.0d0) + (1.0d0 - x_46im)
    else
        tmp = x_46re_m * (x_46im * (-27.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.8e-20) {
		tmp = -1.0 + (1.0 - x_46_im);
	} else {
		tmp = x_46_re_m * (x_46_im * -27.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 3.8e-20:
		tmp = -1.0 + (1.0 - x_46_im)
	else:
		tmp = x_46_re_m * (x_46_im * -27.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.8e-20)
		tmp = Float64(-1.0 + Float64(1.0 - x_46_im));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im * -27.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.8e-20)
		tmp = -1.0 + (1.0 - x_46_im);
	else
		tmp = x_46_re_m * (x_46_im * -27.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 3.8e-20], N[(-1.0 + N[(1.0 - x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im * -27.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.7999999999999998e-20

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. *-un-lft-identity 87.7%

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

      3. distribute-lft-in 87.7%

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

      4. distribute-rgt-out 87.7%

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

      5. metadata-eval 87.7%

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

   4. Applied egg-rr 87.7%

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

      1. pow1 87.7%

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

      2. *-commutative 87.7%

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

      3. add-log-exp 56.7%

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

      4. exp-lft-sqr 56.7%

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

      5. exp-sum 56.7%

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

      6. add-log-exp 87.7%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 54.9%

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

   9. Taylor expanded in x.re around 0 3.4%

      \[\leadsto \color{blue}{-1 \cdot x.im} \]
   10. Step-by-step derivation

       1. neg-mul-1 3.4%

          \[\leadsto \color{blue}{-x.im} \]

   11. Simplified 3.4%

       \[\leadsto \color{blue}{-x.im} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 3.4%

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

       2. expm1-undefine 22.3%

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

   13. Applied egg-rr 22.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-x.im\right)} - 1} \]
   14. Step-by-step derivation

       1. sub-neg 22.3%

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

       2. metadata-eval 22.3%

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

       3. +-commutative 22.3%

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

       4. log1p-undefine 22.3%

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

       5. rem-exp-log 22.3%

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

       6. unsub-neg 22.3%

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

   15. Simplified 22.3%

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

   if 3.7999999999999998e-20 < x.im

   1. Initial program 69.2%

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

      1. difference-of-squares 74.5%

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

   4. Applied egg-rr 74.5%

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

   6. Simplified 31.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Taylor expanded in x.re around 0 34.3%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   8. Taylor expanded in x.im around 0 15.0%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]
   9. Step-by-step derivation

      1. *-commutative 15.0%

         \[\leadsto -27 \cdot \color{blue}{\left(x.re \cdot x.im\right)} \]

      2. *-commutative 15.0%

         \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot -27} \]

      3. associate-*l* 15.0%

         \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot -27\right)} \]

   10. Simplified 15.0%

       \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot -27\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m)));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46im * (x_46re_m * ((x_46im * (-3.0d0)) - x_46re_m)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m)));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m)))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_im * Float64(x_46_re_m * Float64(Float64(x_46_im * -3.0) - x_46_re_m))))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (x_46_im * (x_46_re_m * ((x_46_im * -3.0) - x_46_re_m)));
end

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 85.5%

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

   2. *-commutative 85.5%

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

4. Applied egg-rr 85.5%

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

5. Taylor expanded in x.re around -inf 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in x.im around inf 48.7%

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

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

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

    1. associate--l+ 53.4%

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

    2. distribute-rgt-out-- 53.4%

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

    3. unpow2 53.4%

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

    4. metadata-eval 53.4%

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

    5. associate-*r* 53.4%

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

    6. distribute-rgt-in 51.8%

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

    7. mul-1-neg 51.8%

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

    8. distribute-rgt-neg-in 51.8%

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

    9. associate-*r* 56.9%

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

    10. associate-*r* 63.5%

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

    11. *-commutative 63.5%

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

    12. associate-*r* 63.5%

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

    13. *-commutative 63.5%

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

    14. associate-*r* 63.5%

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

    15. metadata-eval 63.5%

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

    16. distribute-rgt-out-- 63.5%

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

    17. distribute-lft-in 65.1%

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

11. Simplified 67.8%

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

12. Final simplification 67.8%

    \[\leadsto x.im \cdot \left(x.re \cdot \left(x.im \cdot -3 - x.re\right)\right) \]
13. Add Preprocessing

Alternative 7: 51.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-3.0 * (x_46_re_m * (x_46_im * x_46_im)));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((-3.0d0) * (x_46re_m * (x_46im * x_46im)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-3.0 * (x_46_re_m * (x_46_im * x_46_im)));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (-3.0 * (x_46_re_m * (x_46_im * x_46_im)))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * x_46_im))))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (-3.0 * (x_46_re_m * (x_46_im * x_46_im)));
end

Wolfram

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

Derivation

1. Initial program 83.5%

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

3. Simplified 81.6%

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

   1. +-commutative 81.6%

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

   2. associate-*r* 88.1%

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

   3. fma-define 90.4%

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

6. Applied egg-rr 90.4%

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

7. Taylor expanded in x.re around 0 49.1%

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

   1. pow2 49.1%

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

9. Applied egg-rr 49.1%

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

10. Final simplification 49.1%

    \[\leadsto -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \]
11. Add Preprocessing

Alternative 8: 20.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot \left(x.im \cdot -27\right)\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* x.re_m (* x.im -27.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (x_46_im * -27.0));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m * (x_46im * (-27.0d0)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (x_46_im * -27.0));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (x_46_re_m * (x_46_im * -27.0))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(x_46_im * -27.0)))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (x_46_re_m * (x_46_im * -27.0));
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(x$46$re$95$m * N[(x$46$im * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 85.5%

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

4. Applied egg-rr 85.5%

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

6. Simplified 52.9%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.re around 0 28.8%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

8. Taylor expanded in x.im around 0 22.5%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]
9. Step-by-step derivation

   1. *-commutative 22.5%

      \[\leadsto -27 \cdot \color{blue}{\left(x.re \cdot x.im\right)} \]

   2. *-commutative 22.5%

      \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot -27} \]

   3. associate-*l* 22.5%

      \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot -27\right)} \]

10. Simplified 22.5%

    \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot -27\right)} \]
11. Add Preprocessing

Alternative 9: 20.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot -27\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* (* x.re_m x.im) -27.0)))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * -27.0);
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46im) * (-27.0d0))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * -27.0);
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * ((x_46_re_m * x_46_im) * -27.0)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_im) * -27.0))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * ((x_46_re_m * x_46_im) * -27.0);
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 85.5%

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

4. Applied egg-rr 85.5%

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

6. Simplified 52.9%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.re around 0 28.8%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

8. Taylor expanded in x.im around 0 22.5%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

9. Final simplification 22.5%

   \[\leadsto \left(x.re \cdot x.im\right) \cdot -27 \]
10. Add Preprocessing

Alternative 10: 3.6% accurate, 9.5× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * -x_46_im;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * -x_46im
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * -x_46_im;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * -x_46_im

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(-x_46_im))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * -x_46_im;
end

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. *-commutative 83.5%

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

   2. *-un-lft-identity 83.5%

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

   3. distribute-lft-in 83.5%

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

   4. distribute-rgt-out 83.5%

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

   5. metadata-eval 83.5%

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

4. Applied egg-rr 83.5%

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

   1. pow1 83.5%

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

   2. *-commutative 83.5%

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

   3. add-log-exp 49.6%

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

   4. exp-lft-sqr 49.6%

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

   5. exp-sum 49.6%

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

   6. add-log-exp 83.5%

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

   7. flip-+ 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

6. Applied egg-rr 0.0%

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

8. Simplified 51.0%

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

9. Taylor expanded in x.re around 0 3.4%

   \[\leadsto \color{blue}{-1 \cdot x.im} \]
10. Step-by-step derivation

    1. neg-mul-1 3.4%

       \[\leadsto \color{blue}{-x.im} \]

11. Simplified 3.4%

    \[\leadsto \color{blue}{-x.im} \]
12. Add Preprocessing

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot 8 \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im) :precision binary64 (* x.re_s 8.0))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * 8.0;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * 8.0d0
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * 8.0;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * 8.0

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * 8.0)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * 8.0;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * 8.0), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Simplified 81.6%

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

   1. flip-+ 25.1%

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

   2. unpow-prod-down 25.0%

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

   3. div-sub 25.0%

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

   4. pow2 25.0%

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

   5. pow-pow 25.0%

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

   6. metadata-eval 25.0%

      \[\leadsto \frac{{x.re}^{\color{blue}{6}}}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   7. *-commutative 25.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot x.re}} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   8. associate-*r* 25.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot -3\right)} \cdot x.re} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   9. associate-*l* 25.0%

      \[\leadsto \frac{{x.re}^{6}}{{x.re}^{3} - \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-3 \cdot x.re\right)}} - \frac{\left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right) \cdot \left(x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\right)}{{x.re}^{3} - x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   10. pow2 25.0%

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

6. Applied egg-rr 17.5%

   \[\leadsto \color{blue}{\frac{{x.re}^{6}}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)} - \frac{{x.re}^{2} \cdot \left({x.im}^{4} \cdot 9\right)}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)}} \]

8. Simplified 2.8%

   \[\leadsto \color{blue}{8} \]
9. Add Preprocessing

Developer Target 1: 86.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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