math.cube on complex, real part

Percentage Accurate: 83.3% → 99.8%

Time: 7.5s

Alternatives: 9

Speedup: 1.4×

• 43.4% 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: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+95}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m - x.re\_m\right) - x.im \cdot \left(x.re\_m + x.re\_m \cdot 2\right)\right) + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+95)
    (+
     (*
      x.im
      (- (* x.re_m (- x.re_m x.re_m)) (* x.im (+ x.re_m (* x.re_m 2.0)))))
     (pow x.re_m 3.0))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+95) {
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2d+95) then
        tmp = (x_46im * ((x_46re_m * (x_46re_m - x_46re_m)) - (x_46im * (x_46re_m + (x_46re_m * 2.0d0))))) + (x_46re_m ** 3.0d0)
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+95) {
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2e+95:
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2e+95)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)) - Float64(x_46_im * Float64(x_46_re_m + Float64(x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2e+95)
		tmp = (x_46_im * ((x_46_re_m * (x_46_re_m - x_46_re_m)) - (x_46_im * (x_46_re_m + (x_46_re_m * 2.0))))) + (x_46_re_m ^ 3.0);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+95], N[(N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m + N[(x$46$re$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.00000000000000004e95

   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 87.2%

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

      2. *-commutative 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in x.im around 0 90.5%

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

   if 2.00000000000000004e95 < x.re

   1. Initial program 73.2%

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

      1. *-commutative 73.2%

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

      2. *-commutative 73.2%

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

      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. Simplified 85.4%

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

      1. difference-of-squares 73.2%

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

      2. *-commutative 73.2%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+95}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.re\right) - x.im \cdot \left(x.re + x.re \cdot 2\right)\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re\_m \cdot \left(x.im \cdot -2\right), \left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.5e+74)
    (fma
     x.im
     (* x.re_m (* x.im -2.0))
     (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.5e+74) {
		tmp = fma(x_46_im, (x_46_re_m * (x_46_im * -2.0)), ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.5e+74)
		tmp = fma(x_46_im, Float64(x_46_re_m * Float64(x_46_im * -2.0)), Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.5e+74], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.5e74

   1. Initial program 85.4%

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

      1. difference-of-squares 86.8%

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

      2. *-commutative 86.8%

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

   4. Applied egg-rr 86.8%

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

   5. Taylor expanded in x.re around 0 86.8%

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

      1. associate-*r* 86.8%

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

   7. Simplified 86.8%

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

   8. Taylor expanded in x.re around 0 86.8%

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

      1. distribute-lft-in 82.0%

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

      2. mul-1-neg 82.0%

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

      3. sub-neg 82.0%

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

      4. associate-+r+ 82.0%

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

      5. unpow2 82.0%

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

      6. associate-*r* 82.0%

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

      7. distribute-rgt-in 84.0%

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

      8. +-commutative 84.0%

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

      9. mul-1-neg 84.0%

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

      10. sub-neg 84.0%

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

      11. distribute-lft-out 80.1%

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

      12. associate-*r* 87.3%

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

      13. associate-*r* 87.3%

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

      14. unpow2 87.3%

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

      15. distribute-rgt-out 93.0%

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

   10. Simplified 94.0%

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

       1. sub-neg 94.0%

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

       2. +-commutative 94.0%

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

       3. *-commutative 94.0%

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

       4. *-commutative 94.0%

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

       5. *-commutative 94.0%

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

   12. Applied egg-rr 94.0%

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

       1. +-commutative 94.0%

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

       2. distribute-rgt-neg-in 94.0%

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

       3. fma-define 95.5%

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

       4. distribute-rgt-neg-in 95.5%

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

       5. distribute-rgt-neg-in 95.5%

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

       6. metadata-eval 95.5%

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

   14. Simplified 95.5%

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

   if 1.5e74 < x.re

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      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. Simplified 87.2%

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

      1. difference-of-squares 76.6%

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

      2. *-commutative 76.6%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -2\right), \left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 10^{+50}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       1e+50)
    (-
     (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)))
     (* x.im (* x.re_m (* x.im 2.0))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+50) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 1d+50) then
        tmp = ((x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))) - (x_46im * (x_46re_m * (x_46im * 2.0d0)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+50) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+50:
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 1e+50)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im))) - Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * 2.0))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+50)
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+50], N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.0000000000000001e50

   1. Initial program 95.0%

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

      1. difference-of-squares 95.0%

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

      2. *-commutative 95.0%

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in x.re around 0 95.0%

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

      1. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in x.re around 0 95.0%

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

      1. distribute-lft-in 92.4%

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

      2. mul-1-neg 92.4%

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

      3. sub-neg 92.4%

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

      4. associate-+r+ 92.4%

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

      5. unpow2 92.4%

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

      6. associate-*r* 92.4%

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

      7. distribute-rgt-in 94.4%

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

      8. +-commutative 94.4%

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

      9. mul-1-neg 94.4%

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

      10. sub-neg 94.4%

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

      11. distribute-lft-out 90.5%

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

      12. associate-*r* 95.3%

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

      13. associate-*r* 95.3%

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

      14. unpow2 95.3%

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

      15. distribute-rgt-out 99.7%

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

   10. Simplified 99.7%

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

   if 1.0000000000000001e50 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      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. Simplified 69.8%

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

      1. difference-of-squares 69.0%

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

      2. *-commutative 69.0%

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

   8. Applied egg-rr 84.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+50}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ t_1 := x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot 2\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot x.im\right) - t\_1\\ \mathbf{elif}\;x.re\_m \leq 1.5 \cdot 10^{+74}:\\ \;\;\;\;t\_0 - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))
        (t_1 (* x.im (* x.re_m (* x.im 2.0)))))
   (*
    x.re_s
    (if (<= x.re_m 1.2e-139)
      (- (* (- x.re_m x.im) (* x.re_m x.im)) t_1)
      (if (<= x.re_m 1.5e+74) (- t_0 t_1) t_0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double t_1 = x_46_im * (x_46_re_m * (x_46_im * 2.0));
	double tmp;
	if (x_46_re_m <= 1.2e-139) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - t_1;
	} else if (x_46_re_m <= 1.5e+74) {
		tmp = t_0 - t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    t_1 = x_46im * (x_46re_m * (x_46im * 2.0d0))
    if (x_46re_m <= 1.2d-139) then
        tmp = ((x_46re_m - x_46im) * (x_46re_m * x_46im)) - t_1
    else if (x_46re_m <= 1.5d+74) then
        tmp = t_0 - t_1
    else
        tmp = t_0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double t_1 = x_46_im * (x_46_re_m * (x_46_im * 2.0));
	double tmp;
	if (x_46_re_m <= 1.2e-139) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - t_1;
	} else if (x_46_re_m <= 1.5e+74) {
		tmp = t_0 - t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	t_1 = x_46_im * (x_46_re_m * (x_46_im * 2.0))
	tmp = 0
	if x_46_re_m <= 1.2e-139:
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - t_1
	elif x_46_re_m <= 1.5e+74:
		tmp = t_0 - t_1
	else:
		tmp = t_0
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)))
	t_1 = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * 2.0)))
	tmp = 0.0
	if (x_46_re_m <= 1.2e-139)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * x_46_im)) - t_1);
	elseif (x_46_re_m <= 1.5e+74)
		tmp = Float64(t_0 - t_1);
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	t_1 = x_46_im * (x_46_re_m * (x_46_im * 2.0));
	tmp = 0.0;
	if (x_46_re_m <= 1.2e-139)
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - t_1;
	elseif (x_46_re_m <= 1.5e+74)
		tmp = t_0 - t_1;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.2e-139], N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.5e+74], N[(t$95$0 - t$95$1), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < 1.20000000000000007e-139

   1. Initial program 83.2%

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

      1. difference-of-squares 84.9%

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

      2. *-commutative 84.9%

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

   4. Applied egg-rr 84.9%

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

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

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

      1. associate-*r* 84.9%

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

   7. Simplified 84.9%

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

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

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

      1. distribute-lft-in 79.6%

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

      2. mul-1-neg 79.6%

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

      3. sub-neg 79.6%

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

      4. associate-+r+ 79.6%

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

      5. unpow2 79.6%

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

      6. associate-*r* 79.6%

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

      7. distribute-rgt-in 81.4%

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

      8. +-commutative 81.4%

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

      9. mul-1-neg 81.4%

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

      10. sub-neg 81.4%

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

      11. distribute-lft-out 76.7%

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

      12. associate-*r* 84.4%

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

      13. associate-*r* 84.4%

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

      14. unpow2 84.4%

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

      15. distribute-rgt-out 91.5%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in x.re around 0 62.2%

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

   13. Simplified 62.2%

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

   if 1.20000000000000007e-139 < x.re < 1.5e74

   1. Initial program 95.0%

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

      1. difference-of-squares 95.0%

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

      2. *-commutative 95.0%

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in x.re around 0 95.0%

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

      1. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

   if 1.5e74 < x.re

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      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. Simplified 87.2%

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

      1. difference-of-squares 76.6%

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

      2. *-commutative 76.6%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{elif}\;x.re \leq 1.5 \cdot 10^{+74}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.75 \cdot 10^{-102}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.75e-102)
    (- (* (- x.re_m x.im) (* x.re_m x.im)) (* x.im (* x.re_m (* x.im 2.0))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.75e-102) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.75d-102) then
        tmp = ((x_46re_m - x_46im) * (x_46re_m * x_46im)) - (x_46im * (x_46re_m * (x_46im * 2.0d0)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.75e-102) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.75e-102:
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.75e-102)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * x_46_im)) - Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * 2.0))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.75e-102)
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.75e-102], N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.7499999999999999e-102

   1. Initial program 83.8%

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

      1. difference-of-squares 85.5%

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

      2. *-commutative 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in x.re around 0 85.5%

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

      1. associate-*r* 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in x.re around 0 85.5%

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

      1. distribute-lft-in 80.5%

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

      2. mul-1-neg 80.5%

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

      3. sub-neg 80.5%

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

      4. associate-+r+ 80.5%

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

      5. unpow2 80.5%

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

      6. associate-*r* 80.5%

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

      7. distribute-rgt-in 82.1%

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

      8. +-commutative 82.1%

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

      9. mul-1-neg 82.1%

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

      10. sub-neg 82.1%

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

      11. distribute-lft-out 77.6%

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

      12. associate-*r* 85.0%

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

      13. associate-*r* 85.0%

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

      14. unpow2 85.0%

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

      15. distribute-rgt-out 91.8%

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

   10. Simplified 93.0%

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

   11. Taylor expanded in x.re around 0 63.7%

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

   13. Simplified 63.7%

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

   if 2.7499999999999999e-102 < x.re

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. *-commutative 83.6%

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

      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. Simplified 82.7%

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

      1. difference-of-squares 83.6%

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

      2. *-commutative 83.6%

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

   8. Applied egg-rr 90.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.75 \cdot 10^{-102}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 8 \cdot 10^{-103}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 8e-103)
    (* (* x.re_m x.im) (* x.im -3.0))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 8e-103) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 8d-103) then
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 8e-103) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 8e-103:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 8e-103)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 8e-103)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 8e-103], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.99999999999999966e-103

   1. Initial program 83.8%

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

   3. Taylor expanded in x.im around inf 54.1%

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

   5. Applied egg-rr 25.0%

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

      1. exp-sum 25.0%

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

      2. rem-exp-log 29.9%

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

      3. *-commutative 29.9%

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

      4. rem-exp-log 61.6%

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

      5. *-commutative 61.6%

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

   7. Simplified 61.6%

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

   if 7.99999999999999966e-103 < x.re

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. *-commutative 83.6%

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

      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. Simplified 82.7%

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

      1. difference-of-squares 83.6%

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

      2. *-commutative 83.6%

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

   8. Applied egg-rr 90.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8 \cdot 10^{-103}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in x.im around inf 44.3%

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

5. Applied egg-rr 17.3%

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

   1. exp-sum 17.3%

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

   2. rem-exp-log 23.1%

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

   3. *-commutative 23.1%

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

   4. rem-exp-log 50.1%

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

   5. *-commutative 50.1%

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

7. Simplified 50.1%

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

8. Final simplification 50.1%

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

Alternative 8: 4.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. difference-of-squares 84.9%

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

   2. *-commutative 84.9%

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

4. Applied egg-rr 84.9%

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

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

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

   1. distribute-lft-in 79.9%

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

   2. mul-1-neg 79.9%

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

   3. sub-neg 79.9%

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

   4. associate-+r+ 79.9%

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

   5. unpow2 79.9%

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

   6. associate-*r* 79.9%

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

   7. distribute-rgt-in 82.2%

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

   8. +-commutative 82.2%

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

   9. mul-1-neg 82.2%

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

   10. sub-neg 82.2%

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

   11. distribute-lft-out 78.3%

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

   12. associate-*r* 84.1%

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

   13. associate-*r* 84.1%

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

   14. unpow2 84.1%

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

   15. distribute-rgt-out 90.0%

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

7. Simplified 90.8%

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

8. Taylor expanded in x.re around inf 63.8%

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

10. Simplified 4.5%

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

11. Taylor expanded in x.re around inf 4.8%

    \[\leadsto \color{blue}{27 \cdot x.re} \]
12. Step-by-step derivation

    1. *-commutative 4.8%

       \[\leadsto \color{blue}{x.re \cdot 27} \]

13. Simplified 4.8%

    \[\leadsto \color{blue}{x.re \cdot 27} \]
14. Add Preprocessing

Alternative 9: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. difference-of-squares 84.9%

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

   2. *-commutative 84.9%

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

4. Applied egg-rr 84.9%

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

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

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

   1. distribute-lft-in 79.9%

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

   2. mul-1-neg 79.9%

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

   3. sub-neg 79.9%

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

   4. associate-+r+ 79.9%

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

   5. unpow2 79.9%

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

   6. associate-*r* 79.9%

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

   7. distribute-rgt-in 82.2%

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

   8. +-commutative 82.2%

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

   9. mul-1-neg 82.2%

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

   10. sub-neg 82.2%

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

   11. distribute-lft-out 78.3%

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

   12. associate-*r* 84.1%

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

   13. associate-*r* 84.1%

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

   14. unpow2 84.1%

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

   15. distribute-rgt-out 90.0%

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

7. Simplified 90.8%

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

8. Taylor expanded in x.re around inf 63.8%

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

10. Simplified 4.5%

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

11. Taylor expanded in x.re around 0 2.8%

    \[\leadsto \color{blue}{27} \]
12. Add Preprocessing

Developer target: 87.5% 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 2024108 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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