math.cube on complex, real part

Percentage Accurate: 82.6% → 99.8%

Time: 10.5s

Alternatives: 9

Speedup: 0.9×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\left({x.re\_m}^{3} - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re\_m}{x.im\_m}\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+99)
    (-
     (- (pow x.re_m 3.0) (* x.im_m (* x.re_m x.im_m)))
     (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
    (* (* x.im_m (* x.re_m (+ x.re_m -27.0))) (+ -1.0 (/ x.re_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2e+99) {
		tmp = (pow(x_46_re_m, 3.0) - (x_46_im_m * (x_46_re_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2d+99) then
        tmp = ((x_46re_m ** 3.0d0) - (x_46im_m * (x_46re_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))
    else
        tmp = (x_46im_m * (x_46re_m * (x_46re_m + (-27.0d0)))) * ((-1.0d0) + (x_46re_m / x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2e+99) {
		tmp = (Math.pow(x_46_re_m, 3.0) - (x_46_im_m * (x_46_re_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 2e+99:
		tmp = (math.pow(x_46_re_m, 3.0) - (x_46_im_m * (x_46_re_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))
	else:
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 2e+99)
		tmp = Float64(Float64((x_46_re_m ^ 3.0) - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + -27.0))) * Float64(-1.0 + Float64(x_46_re_m / x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 2e+99)
		tmp = ((x_46_re_m ^ 3.0) - (x_46_im_m * (x_46_re_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	else
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+99], N[(N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.9999999999999999e99

   1. Initial program 87.0%

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

      1. difference-of-squares 89.4%

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

      2. *-commutative 89.4%

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

   4. Applied egg-rr 89.4%

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

   5. Taylor expanded in x.im around 0 91.8%

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

   6. Taylor expanded in x.im around inf 91.8%

      \[\leadsto \left(x.im \cdot \color{blue}{\left(-1 \cdot \left(x.im \cdot x.re\right)\right)} + {x.re}^{3}\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg 91.8%

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

      2. distribute-lft-neg-out 91.8%

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

      3. *-commutative 91.8%

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

   8. Simplified 91.8%

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

   if 1.9999999999999999e99 < x.re

   1. Initial program 55.6%

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

      1. difference-of-squares 68.9%

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

   4. Applied egg-rr 68.9%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in x.im around inf 51.4%

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

      1. sub-neg 51.4%

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

      2. metadata-eval 51.4%

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

      3. remove-double-neg 51.4%

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

      4. mul-1-neg 51.4%

         \[\leadsto \left(\left(-\color{blue}{-1 \cdot x.im}\right) \cdot \left(\left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right) + -27\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. +-commutative 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(-27 + \left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. associate-+l+ 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(\left(-27 + x.re\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      7. +-commutative 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right)} + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      8. distribute-lft-neg-in 51.4%

         \[\leadsto \color{blue}{\left(-\left(-1 \cdot x.im\right) \cdot \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      9. distribute-rgt-neg-in 51.4%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot x.im\right) \cdot \left(-\left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. mul-1-neg 51.4%

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

      11. neg-sub0 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(0 - \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. associate--l- 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(\left(0 - \left(x.re + -27\right)\right) - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. neg-sub0 51.4%

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

      14. mul-1-neg 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{-1 \cdot \left(x.re + -27\right)} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. *-commutative 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right) \cdot -1} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. sub-neg 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. metadata-eval 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \left(x.re + \color{blue}{-27}\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. *-commutative 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{\color{blue}{\left(x.re + -27\right) \cdot x.re}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. associate-/l* 53.6%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \color{blue}{\left(x.re + -27\right) \cdot \frac{x.re}{x.im}}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   9. Simplified 53.6%

      \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   10. Step-by-step derivation

       1. sub-neg 53.6%

          \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]

       2. *-commutative 53.6%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]

   11. Applied egg-rr 91.1%

       \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right) + 0} \]
   12. Step-by-step derivation

       1. +-rgt-identity 91.1%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right)} \]

       2. associate-*r* 91.1%

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)} \]

       3. metadata-eval 91.1%

          \[\leadsto \left(x.re \cdot \left(x.re + \color{blue}{\left(-27\right)}\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       4. sub-neg 91.1%

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.re - 27\right)}\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       5. *-commutative 91.1%

          \[\leadsto \left(x.re \cdot \left(x.re - 27\right)\right) \cdot \color{blue}{\left(x.im \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)} \]

       6. associate-*r* 91.1%

          \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(x.re - 27\right)\right) \cdot x.im\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]

       7. *-commutative 91.1%

          \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\right)} \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       8. sub-neg 91.1%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       9. metadata-eval 91.1%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

   13. Simplified 91.1%

       \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\left({x.re}^{3} - x.im \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+99}:\\ \;\;\;\;{x.re\_m}^{3} + x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re\_m}{x.im\_m}\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+99)
    (+ (pow x.re_m 3.0) (* x.im_m (* x.re_m (* x.im_m -3.0))))
    (* (* x.im_m (* x.re_m (+ x.re_m -27.0))) (+ -1.0 (/ x.re_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2e+99) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_im_m * (x_46_re_m * (x_46_im_m * -3.0)));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2d+99) then
        tmp = (x_46re_m ** 3.0d0) + (x_46im_m * (x_46re_m * (x_46im_m * (-3.0d0))))
    else
        tmp = (x_46im_m * (x_46re_m * (x_46re_m + (-27.0d0)))) * ((-1.0d0) + (x_46re_m / x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2e+99) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_im_m * (x_46_re_m * (x_46_im_m * -3.0)));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 2e+99:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_im_m * (x_46_re_m * (x_46_im_m * -3.0)))
	else:
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 2e+99)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * -3.0))));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + -27.0))) * Float64(-1.0 + Float64(x_46_re_m / x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 2e+99)
		tmp = (x_46_re_m ^ 3.0) + (x_46_im_m * (x_46_re_m * (x_46_im_m * -3.0)));
	else
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+99], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.9999999999999999e99

   1. Initial program 87.0%

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

   3. Simplified 85.6%

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

      1. associate-*r* 85.6%

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

      2. associate-*l* 85.6%

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

      3. +-commutative 85.6%

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

      4. associate-*r* 91.8%

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

      5. associate-*r* 91.8%

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

      6. fma-define 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. fma-undefine 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*l* 91.8%

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

   8. Applied egg-rr 91.8%

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

   if 1.9999999999999999e99 < x.re

   1. Initial program 55.6%

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

      1. difference-of-squares 68.9%

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

   4. Applied egg-rr 68.9%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in x.im around inf 51.4%

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

      1. sub-neg 51.4%

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

      2. metadata-eval 51.4%

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

      3. remove-double-neg 51.4%

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

      4. mul-1-neg 51.4%

         \[\leadsto \left(\left(-\color{blue}{-1 \cdot x.im}\right) \cdot \left(\left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right) + -27\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. +-commutative 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(-27 + \left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. associate-+l+ 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(\left(-27 + x.re\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      7. +-commutative 51.4%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right)} + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      8. distribute-lft-neg-in 51.4%

         \[\leadsto \color{blue}{\left(-\left(-1 \cdot x.im\right) \cdot \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      9. distribute-rgt-neg-in 51.4%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot x.im\right) \cdot \left(-\left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. mul-1-neg 51.4%

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

      11. neg-sub0 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(0 - \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. associate--l- 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(\left(0 - \left(x.re + -27\right)\right) - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. neg-sub0 51.4%

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

      14. mul-1-neg 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{-1 \cdot \left(x.re + -27\right)} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. *-commutative 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right) \cdot -1} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. sub-neg 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. metadata-eval 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \left(x.re + \color{blue}{-27}\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. *-commutative 51.4%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{\color{blue}{\left(x.re + -27\right) \cdot x.re}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. associate-/l* 53.6%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \color{blue}{\left(x.re + -27\right) \cdot \frac{x.re}{x.im}}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   9. Simplified 53.6%

      \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   10. Step-by-step derivation

       1. sub-neg 53.6%

          \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]

       2. *-commutative 53.6%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]

   11. Applied egg-rr 91.1%

       \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right) + 0} \]
   12. Step-by-step derivation

       1. +-rgt-identity 91.1%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right)} \]

       2. associate-*r* 91.1%

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)} \]

       3. metadata-eval 91.1%

          \[\leadsto \left(x.re \cdot \left(x.re + \color{blue}{\left(-27\right)}\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       4. sub-neg 91.1%

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.re - 27\right)}\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       5. *-commutative 91.1%

          \[\leadsto \left(x.re \cdot \left(x.re - 27\right)\right) \cdot \color{blue}{\left(x.im \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)} \]

       6. associate-*r* 91.1%

          \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(x.re - 27\right)\right) \cdot x.im\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]

       7. *-commutative 91.1%

          \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\right)} \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       8. sub-neg 91.1%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       9. metadata-eval 91.1%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

   13. Simplified 91.1%

       \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+99}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 0.79:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re\_m}{x.im\_m}\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 0.79)
    (* x.im_m (* x.re_m (* x.im_m -2.0)))
    (if (<= x.re_m 6.5e+91)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))) 0.5)
      (* (* x.im_m (* x.re_m (+ x.re_m -27.0))) (+ -1.0 (/ x.re_m x.im_m)))))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 0.79) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	} else if (x_46_re_m <= 6.5e+91) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 0.79d0) then
        tmp = x_46im_m * (x_46re_m * (x_46im_m * (-2.0d0)))
    else if (x_46re_m <= 6.5d+91) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - 0.5d0
    else
        tmp = (x_46im_m * (x_46re_m * (x_46re_m + (-27.0d0)))) * ((-1.0d0) + (x_46re_m / x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 0.79) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	} else if (x_46_re_m <= 6.5e+91) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 0.79:
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0))
	elif x_46_re_m <= 6.5e+91:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5
	else:
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 0.79)
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * -2.0)));
	elseif (x_46_re_m <= 6.5e+91)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - 0.5);
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + -27.0))) * Float64(-1.0 + Float64(x_46_re_m / x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 0.79)
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	elseif (x_46_re_m <= 6.5e+91)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	else
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 0.79], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 6.5e+91], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 0.79000000000000004

   1. Initial program 85.8%

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

      1. difference-of-squares 88.4%

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

   4. Applied egg-rr 88.4%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in x.re around 0 34.1%

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

      1. associate-*r* 34.1%

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

      2. *-commutative 34.1%

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

   9. Simplified 34.1%

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

   10. Taylor expanded in x.im around 0 36.2%

       \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re + -2 \cdot \left(x.im \cdot x.re\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 36.2%

          \[\leadsto x.im \cdot \left(-27 \cdot x.re + \color{blue}{\left(-2 \cdot x.im\right) \cdot x.re}\right) \]

       2. distribute-rgt-out 36.2%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(-27 + -2 \cdot x.im\right)\right)} \]

       3. *-commutative 36.2%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(-27 + \color{blue}{x.im \cdot -2}\right)\right) \]

   12. Simplified 36.2%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

   13. Taylor expanded in x.im around inf 40.5%

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

       1. *-commutative 40.5%

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

       2. *-commutative 40.5%

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

       3. associate-*r* 40.5%

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

   15. Simplified 40.5%

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

   if 0.79000000000000004 < x.re < 6.4999999999999997e91

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 79.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.5} \]

   if 6.4999999999999997e91 < x.re

   1. Initial program 57.4%

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

      1. difference-of-squares 70.2%

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

   4. Applied egg-rr 70.2%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in x.im around inf 53.5%

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

      1. sub-neg 53.5%

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

      2. metadata-eval 53.5%

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

      3. remove-double-neg 53.5%

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

      4. mul-1-neg 53.5%

         \[\leadsto \left(\left(-\color{blue}{-1 \cdot x.im}\right) \cdot \left(\left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right) + -27\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. +-commutative 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(-27 + \left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. associate-+l+ 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(\left(-27 + x.re\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      7. +-commutative 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right)} + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      8. distribute-lft-neg-in 53.5%

         \[\leadsto \color{blue}{\left(-\left(-1 \cdot x.im\right) \cdot \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      9. distribute-rgt-neg-in 53.5%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot x.im\right) \cdot \left(-\left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. mul-1-neg 53.5%

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

      11. neg-sub0 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(0 - \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. associate--l- 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(\left(0 - \left(x.re + -27\right)\right) - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. neg-sub0 53.5%

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

      14. mul-1-neg 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{-1 \cdot \left(x.re + -27\right)} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. *-commutative 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right) \cdot -1} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. sub-neg 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. metadata-eval 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \left(x.re + \color{blue}{-27}\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. *-commutative 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{\color{blue}{\left(x.re + -27\right) \cdot x.re}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. associate-/l* 55.6%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \color{blue}{\left(x.re + -27\right) \cdot \frac{x.re}{x.im}}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   9. Simplified 55.6%

      \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   10. Step-by-step derivation

       1. sub-neg 55.6%

          \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]

       2. *-commutative 55.6%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]

   11. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right) + 0} \]
   12. Step-by-step derivation

       1. +-rgt-identity 89.4%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right)} \]

       2. associate-*r* 89.4%

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)} \]

       3. metadata-eval 89.4%

          \[\leadsto \left(x.re \cdot \left(x.re + \color{blue}{\left(-27\right)}\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       4. sub-neg 89.4%

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.re - 27\right)}\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       5. *-commutative 89.4%

          \[\leadsto \left(x.re \cdot \left(x.re - 27\right)\right) \cdot \color{blue}{\left(x.im \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)} \]

       6. associate-*r* 89.4%

          \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(x.re - 27\right)\right) \cdot x.im\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]

       7. *-commutative 89.4%

          \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\right)} \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       8. sub-neg 89.4%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       9. metadata-eval 89.4%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

   13. Simplified 89.4%

       \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 0.79:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -2\right)\right)\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re\_m}{x.im\_m}\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 6.5e+91)
    (-
     (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
     (* x.im_m (* (* x.re_m x.im_m) 2.0)))
    (* (* x.im_m (* x.re_m (+ x.re_m -27.0))) (+ -1.0 (/ x.re_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 6.5e+91) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 6.5d+91) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) * 2.0d0))
    else
        tmp = (x_46im_m * (x_46re_m * (x_46re_m + (-27.0d0)))) * ((-1.0d0) + (x_46re_m / x_46im_m))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 6.5e+91) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 6.5e+91:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0))
	else:
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 6.5e+91)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + -27.0))) * Float64(-1.0 + Float64(x_46_re_m / x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 6.5e+91)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	else
		tmp = (x_46_im_m * (x_46_re_m * (x_46_re_m + -27.0))) * (-1.0 + (x_46_re_m / x_46_im_m));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 6.5e+91], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.4999999999999997e91

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. *-un-lft-identity 86.9%

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

      3. *-un-lft-identity 86.9%

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

      4. distribute-rgt-out 86.9%

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

      5. metadata-eval 86.9%

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

   4. Applied egg-rr 86.9%

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

   if 6.4999999999999997e91 < x.re

   1. Initial program 57.4%

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

      1. difference-of-squares 70.2%

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

   4. Applied egg-rr 70.2%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in x.im around inf 53.5%

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

      1. sub-neg 53.5%

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

      2. metadata-eval 53.5%

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

      3. remove-double-neg 53.5%

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

      4. mul-1-neg 53.5%

         \[\leadsto \left(\left(-\color{blue}{-1 \cdot x.im}\right) \cdot \left(\left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right) + -27\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. +-commutative 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(-27 + \left(x.re + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. associate-+l+ 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \color{blue}{\left(\left(-27 + x.re\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      7. +-commutative 53.5%

         \[\leadsto \left(\left(--1 \cdot x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right)} + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      8. distribute-lft-neg-in 53.5%

         \[\leadsto \color{blue}{\left(-\left(-1 \cdot x.im\right) \cdot \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      9. distribute-rgt-neg-in 53.5%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot x.im\right) \cdot \left(-\left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. mul-1-neg 53.5%

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

      11. neg-sub0 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(0 - \left(\left(x.re + -27\right) + \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. associate--l- 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \color{blue}{\left(\left(0 - \left(x.re + -27\right)\right) - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. neg-sub0 53.5%

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

      14. mul-1-neg 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{-1 \cdot \left(x.re + -27\right)} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      15. *-commutative 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\color{blue}{\left(x.re + -27\right) \cdot -1} - \frac{x.re \cdot \left(x.re - 27\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      16. sub-neg 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      17. metadata-eval 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{x.re \cdot \left(x.re + \color{blue}{-27}\right)}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      18. *-commutative 53.5%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \frac{\color{blue}{\left(x.re + -27\right) \cdot x.re}}{x.im}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      19. associate-/l* 55.6%

          \[\leadsto \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot -1 - \color{blue}{\left(x.re + -27\right) \cdot \frac{x.re}{x.im}}\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   9. Simplified 55.6%

      \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   10. Step-by-step derivation

       1. sub-neg 55.6%

          \[\leadsto \color{blue}{\left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]

       2. *-commutative 55.6%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(-x.im\right) \cdot \left(\left(x.re + -27\right) \cdot \left(-1 - \frac{x.re}{x.im}\right)\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]

   11. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right) + 0} \]
   12. Step-by-step derivation

       1. +-rgt-identity 89.4%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)\right)} \]

       2. associate-*r* 89.4%

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right)} \]

       3. metadata-eval 89.4%

          \[\leadsto \left(x.re \cdot \left(x.re + \color{blue}{\left(-27\right)}\right)\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       4. sub-neg 89.4%

          \[\leadsto \left(x.re \cdot \color{blue}{\left(x.re - 27\right)}\right) \cdot \left(\left(-1 + \frac{x.re}{x.im}\right) \cdot x.im\right) \]

       5. *-commutative 89.4%

          \[\leadsto \left(x.re \cdot \left(x.re - 27\right)\right) \cdot \color{blue}{\left(x.im \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)} \]

       6. associate-*r* 89.4%

          \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(x.re - 27\right)\right) \cdot x.im\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]

       7. *-commutative 89.4%

          \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\right)} \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       8. sub-neg 89.4%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

       9. metadata-eval 89.4%

          \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) \]

   13. Simplified 89.4%

       \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 0.79:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - 0.5\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 0.79)
    (* x.im_m (* x.re_m (* x.im_m -2.0)))
    (- (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))) 0.5))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 0.79) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 0.79d0) then
        tmp = x_46im_m * (x_46re_m * (x_46im_m * (-2.0d0)))
    else
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - 0.5d0
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 0.79) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 0.79:
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0))
	else:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 0.79)
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * -2.0)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - 0.5);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 0.79)
		tmp = x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0));
	else
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 0.79], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 0.79000000000000004

   1. Initial program 85.8%

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

      1. difference-of-squares 88.4%

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

   4. Applied egg-rr 88.4%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in x.re around 0 34.1%

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

      1. associate-*r* 34.1%

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

      2. *-commutative 34.1%

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

   9. Simplified 34.1%

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

   10. Taylor expanded in x.im around 0 36.2%

       \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re + -2 \cdot \left(x.im \cdot x.re\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 36.2%

          \[\leadsto x.im \cdot \left(-27 \cdot x.re + \color{blue}{\left(-2 \cdot x.im\right) \cdot x.re}\right) \]

       2. distribute-rgt-out 36.2%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(-27 + -2 \cdot x.im\right)\right)} \]

       3. *-commutative 36.2%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(-27 + \color{blue}{x.im \cdot -2}\right)\right) \]

   12. Simplified 36.2%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

   13. Taylor expanded in x.im around inf 40.5%

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

       1. *-commutative 40.5%

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

       2. *-commutative 40.5%

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

       3. associate-*r* 40.5%

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

   15. Simplified 40.5%

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

   if 0.79000000000000004 < x.re

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

      2. *-commutative 68.2%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 80.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 0.79:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 22.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot -27\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (* x.re_s (if (<= x.im_m 5.8e-21) 0.0 (* (* x.re_m x.im_m) -27.0))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_im_m <= 5.8e-21) {
		tmp = 0.0;
	} else {
		tmp = (x_46_re_m * x_46_im_m) * -27.0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5.8d-21) then
        tmp = 0.0d0
    else
        tmp = (x_46re_m * x_46im_m) * (-27.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_im_m <= 5.8e-21) {
		tmp = 0.0;
	} else {
		tmp = (x_46_re_m * x_46_im_m) * -27.0;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_im_m <= 5.8e-21:
		tmp = 0.0
	else:
		tmp = (x_46_re_m * x_46_im_m) * -27.0
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_im_m <= 5.8e-21)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * -27.0);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_im_m <= 5.8e-21)
		tmp = 0.0;
	else
		tmp = (x_46_re_m * x_46_im_m) * -27.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 5.8e-21], 0.0, N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -27.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.8e-21

   1. Initial program 88.6%

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

      1. difference-of-squares 90.8%

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

      2. *-commutative 90.8%

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

   4. Applied egg-rr 90.8%

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

   6. Applied egg-rr 23.5%

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

      1. flip-+ 0.0%

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

      2. +-inverses 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. flip-+ 33.3%

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

      7. *-commutative 33.3%

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

      8. add-sqr-sqrt 9.7%

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

      9. sqrt-unprod 18.7%

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

      10. sqr-neg 18.7%

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

      11. sqrt-unprod 19.1%

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

      12. add-sqr-sqrt 31.4%

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

      13. cancel-sign-sub-inv 31.4%

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

      14. *-commutative 31.4%

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

      15. +-inverses 37.4%

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

      16. neg-sub0 37.4%

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

      17. *-commutative 37.4%

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

      18. distribute-lft-neg-in 37.4%

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

      19. add-sqr-sqrt 25.8%

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

      20. sqrt-unprod 37.0%

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

      21. sqr-neg 37.0%

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

      22. sqrt-unprod 11.7%

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

   8. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{0} \]

   if 5.8e-21 < x.im

   1. Initial program 63.9%

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

      1. difference-of-squares 73.3%

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

   4. Applied egg-rr 73.3%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in x.re around 0 37.5%

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

      1. associate-*r* 37.5%

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

      2. *-commutative 37.5%

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

   9. Simplified 37.5%

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

   10. Taylor expanded in x.im around 0 17.1%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot -27\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 17.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot 2\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (* x.re_s (if (<= x.re_m 1.25e-33) 0.0 (* x.re_m 2.0))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 1.25e-33) {
		tmp = 0.0;
	} else {
		tmp = x_46_re_m * 2.0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1.25d-33) then
        tmp = 0.0d0
    else
        tmp = x_46re_m * 2.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 1.25e-33) {
		tmp = 0.0;
	} else {
		tmp = x_46_re_m * 2.0;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if x_46_re_m <= 1.25e-33:
		tmp = 0.0
	else:
		tmp = x_46_re_m * 2.0
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 1.25e-33)
		tmp = 0.0;
	else
		tmp = Float64(x_46_re_m * 2.0);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (x_46_re_m <= 1.25e-33)
		tmp = 0.0;
	else
		tmp = x_46_re_m * 2.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.25e-33], 0.0, N[(x$46$re$95$m * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.25000000000000007e-33

   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 87.8%

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

      2. *-commutative 87.8%

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

   4. Applied egg-rr 87.8%

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

   6. Applied egg-rr 26.4%

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

      1. flip-+ 0.0%

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

      2. +-inverses 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. flip-+ 35.6%

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

      7. *-commutative 35.6%

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

      8. add-sqr-sqrt 18.5%

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

      9. sqrt-unprod 20.4%

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

      10. sqr-neg 20.4%

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

      11. sqrt-unprod 16.3%

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

      12. add-sqr-sqrt 37.8%

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

      13. cancel-sign-sub-inv 37.8%

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

      14. *-commutative 37.8%

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

      15. +-inverses 41.1%

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

      16. neg-sub0 41.1%

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

      17. *-commutative 41.1%

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

      18. distribute-lft-neg-in 41.1%

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

      19. add-sqr-sqrt 19.2%

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

      20. sqrt-unprod 31.6%

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

      21. sqr-neg 31.6%

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

      22. sqrt-unprod 12.9%

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

   8. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{0} \]

   if 1.25000000000000007e-33 < x.re

   1. Initial program 72.5%

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

      1. difference-of-squares 80.8%

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

      2. *-commutative 80.8%

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

   4. Applied egg-rr 80.8%

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

   6. Applied egg-rr 30.3%

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

   7. Taylor expanded in x.im around 0 4.6%

      \[\leadsto \color{blue}{2 \cdot x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot -2\right)\right)\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (* x.re_s (* x.im_m (* x.re_m (* x.im_m -2.0)))))

C

x.im_m = fabs(x_46_im);
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_m) {
	return x_46_re_s * (x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0)));
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46im_m * (x_46re_m * (x_46im_m * (-2.0d0))))
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	return x_46_re_s * (x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0)));
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	return x_46_re_s * (x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0)))

Julia

x.im_m = abs(x_46_im)
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_m)
	return Float64(x_46_re_s * Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m * -2.0))))
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = x_46_re_s * (x_46_im_m * (x_46_re_m * (x_46_im_m * -2.0)));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

   1. difference-of-squares 85.8%

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

4. Applied egg-rr 85.8%

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

6. Simplified 51.3%

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

7. Taylor expanded in x.re around 0 30.1%

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

   1. associate-*r* 30.0%

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

   2. *-commutative 30.0%

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

9. Simplified 30.0%

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

10. Taylor expanded in x.im around 0 34.3%

    \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re + -2 \cdot \left(x.im \cdot x.re\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r* 34.3%

       \[\leadsto x.im \cdot \left(-27 \cdot x.re + \color{blue}{\left(-2 \cdot x.im\right) \cdot x.re}\right) \]

    2. distribute-rgt-out 34.3%

       \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(-27 + -2 \cdot x.im\right)\right)} \]

    3. *-commutative 34.3%

       \[\leadsto x.im \cdot \left(x.re \cdot \left(-27 + \color{blue}{x.im \cdot -2}\right)\right) \]

12. Simplified 34.3%

    \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

13. Taylor expanded in x.im around inf 37.5%

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

    1. *-commutative 37.5%

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

    2. *-commutative 37.5%

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

    3. associate-*r* 37.5%

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

15. Simplified 37.5%

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

Alternative 9: 15.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot 0 \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m) :precision binary64 (* x.re_s 0.0))

C

x.im_m = fabs(x_46_im);
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_m) {
	return x_46_re_s * 0.0;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * 0.0d0
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	return x_46_re_s * 0.0;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	return x_46_re_s * 0.0

Julia

x.im_m = abs(x_46_im)
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_m)
	return Float64(x_46_re_s * 0.0)
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = x_46_re_s * 0.0;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

   1. difference-of-squares 85.8%

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

   2. *-commutative 85.8%

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

4. Applied egg-rr 85.8%

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

6. Applied egg-rr 27.5%

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

   1. flip-+ 0.0%

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

   2. +-inverses 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 30.5%

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

   7. *-commutative 30.5%

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

   8. add-sqr-sqrt 13.8%

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

   9. sqrt-unprod 15.4%

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

   10. sqr-neg 15.4%

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

   11. sqrt-unprod 13.6%

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

   12. add-sqr-sqrt 29.3%

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

   13. cancel-sign-sub-inv 29.3%

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

   14. *-commutative 29.3%

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

   15. +-inverses 37.5%

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

   16. neg-sub0 37.5%

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

   17. *-commutative 37.5%

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

   18. distribute-lft-neg-in 37.5%

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

   19. add-sqr-sqrt 18.3%

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

   20. sqrt-unprod 31.8%

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

   21. sqr-neg 31.8%

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

   22. sqrt-unprod 13.8%

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

8. Applied egg-rr 13.4%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 99.8% 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 2024144 
(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)))