math.cube on complex, real part

Percentage Accurate: 82.7% → 93.9%

Time: 9.0s

Alternatives: 13

Speedup: 0.9×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.im, x.im \cdot -3, {x.re\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

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 2.15e+96)
    (fma (* x.re_m x.im) (* x.im -3.0) (pow x.re_m 3.0))
    (pow x.re_m 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) {
	double tmp;
	if (x_46_re_m <= 2.15e+96) {
		tmp = fma((x_46_re_m * x_46_im), (x_46_im * -3.0), pow(x_46_re_m, 3.0));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.15e+96)
		tmp = fma(Float64(x_46_re_m * x_46_im), Float64(x_46_im * -3.0), (x_46_re_m ^ 3.0));
	else
		tmp = x_46_re_m ^ 3.0;
	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, 2.15e+96], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.15000000000000001e96

   1. Initial program 85.5%

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

   3. Simplified 85.0%

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

      1. associate-*r* 85.0%

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

      2. associate-*l* 85.0%

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

      3. +-commutative 85.0%

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

      4. associate-*r* 93.3%

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

      5. associate-*r* 93.4%

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

      6. fma-define 94.7%

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

   6. Applied egg-rr 94.7%

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

   if 2.15000000000000001e96 < x.re

   1. Initial program 72.5%

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

   3. Simplified 70.0%

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

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

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;{x.re\_m}^{3} + -3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

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 2.15e+96)
    (+ (pow x.re_m 3.0) (* -3.0 (* x.im (* x.re_m x.im))))
    (pow x.re_m 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) {
	double tmp;
	if (x_46_re_m <= 2.15e+96) {
		tmp = pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.15d+96) then
        tmp = (x_46re_m ** 3.0d0) + ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.15e+96) {
		tmp = Math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.15e+96:
		tmp = math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))
	else:
		tmp = math.pow(x_46_re_m, 3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.15e+96)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.15e+96)
		tmp = (x_46_re_m ^ 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.15e+96], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.15000000000000001e96

   1. Initial program 85.5%

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

   3. Simplified 85.0%

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

      1. add-sqr-sqrt 48.8%

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

      2. pow2 48.8%

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

      3. *-commutative 48.8%

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

      4. sqrt-prod 28.9%

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

      5. sqrt-prod 14.2%

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

      6. add-sqr-sqrt 34.6%

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

   6. Applied egg-rr 34.6%

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

      1. unpow2 34.6%

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

      2. *-commutative 34.6%

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

      3. associate-*r* 34.6%

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

      4. associate-*r* 34.5%

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

      5. add-sqr-sqrt 93.3%

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

   8. Applied egg-rr 93.3%

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

   if 2.15000000000000001e96 < x.re

   1. Initial program 72.5%

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

   3. Simplified 70.0%

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

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

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.9 \cdot 10^{+157}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re\_m \cdot \frac{x.re\_m + -19683}{x.im}\right)\\ \end{array} \end{array} \]

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 1.9e+157)
    (-
     (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))
     (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
    (* x.im (+ -2.0 (* x.re_m (/ (+ x.re_m -19683.0) 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 <= 1.9e+157) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1.9d+157) then
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else
        tmp = x_46im * ((-2.0d0) + (x_46re_m * ((x_46re_m + (-19683.0d0)) / x_46im)))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.9e+157) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1.9e+157:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 1.9e+157)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(-2.0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m + -19683.0) / x_46_im))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1.9e+157)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	else
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.9e+157], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-2.0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + -19683.0), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.9e157

   1. Initial program 86.2%

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

      1. difference-of-squares 88.0%

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

      2. *-commutative 88.0%

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

   4. Applied egg-rr 88.0%

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

   if 1.9e157 < x.re

   1. Initial program 62.1%

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

      1. difference-of-squares 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in x.im around inf 27.6%

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

      1. unpow2 27.6%

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

      2. associate-*l* 27.6%

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

      3. associate-*r/ 27.6%

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

      4. distribute-lft-out 62.1%

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

      5. +-commutative 62.1%

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

      6. remove-double-neg 62.1%

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

      7. mul-1-neg 62.1%

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

      8. sub-neg 62.1%

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

      9. *-commutative 62.1%

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

      10. associate-/l* 62.1%

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

      11. *-commutative 62.1%

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

      12. distribute-lft-out-- 62.1%

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

      13. sub-neg 62.1%

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

      14. metadata-eval 62.1%

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

      15. sub-neg 62.1%

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

      16. metadata-eval 62.1%

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

      17. +-commutative 62.1%

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

   9. Simplified 62.1%

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

   10. Taylor expanded in x.im around 0 62.1%

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

   12. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{x.im \cdot \left({x.re}^{2} \cdot \frac{x.re + -27}{x.im}\right) + x.im \cdot \frac{0}{0}} \]

   14. Simplified 96.6%

       \[\leadsto \color{blue}{x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.9 \cdot 10^{+157}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re\_m \cdot \frac{x.re\_m + -19683}{x.im}\right)\\ \end{array} \end{array} \]

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 1.35e+154)
    (-
     (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (* x.im (+ -2.0 (* x.re_m (/ (+ x.re_m -19683.0) 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 <= 1.35e+154) {
		tmp = (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) * 2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = x_46im * ((-2.0d0) + (x_46re_m * ((x_46re_m + (-19683.0d0)) / x_46im)))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.35e+154) {
		tmp = (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) * 2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1.35e+154:
		tmp = (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) * 2.0))
	else:
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 1.35e+154)
		tmp = 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) * 2.0)));
	else
		tmp = Float64(x_46_im * Float64(-2.0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m + -19683.0) / x_46_im))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1.35e+154)
		tmp = (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) * 2.0));
	else
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.35e+154], 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] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-2.0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + -19683.0), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.35000000000000003e154

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. *-un-lft-identity 86.2%

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

      3. *-un-lft-identity 86.2%

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

      4. distribute-rgt-out 86.2%

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

      5. metadata-eval 86.2%

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

   4. Applied egg-rr 86.2%

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

   if 1.35000000000000003e154 < x.re

   1. Initial program 62.1%

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

      1. difference-of-squares 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in x.im around inf 27.6%

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

      1. unpow2 27.6%

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

      2. associate-*l* 27.6%

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

      3. associate-*r/ 27.6%

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

      4. distribute-lft-out 62.1%

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

      5. +-commutative 62.1%

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

      6. remove-double-neg 62.1%

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

      7. mul-1-neg 62.1%

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

      8. sub-neg 62.1%

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

      9. *-commutative 62.1%

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

      10. associate-/l* 62.1%

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

      11. *-commutative 62.1%

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

      12. distribute-lft-out-- 62.1%

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

      13. sub-neg 62.1%

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

      14. metadata-eval 62.1%

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

      15. sub-neg 62.1%

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

      16. metadata-eval 62.1%

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

      17. +-commutative 62.1%

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

   9. Simplified 62.1%

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

   10. Taylor expanded in x.im around 0 62.1%

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

   12. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{x.im \cdot \left({x.re}^{2} \cdot \frac{x.re + -27}{x.im}\right) + x.im \cdot \frac{0}{0}} \]

   14. Simplified 96.6%

       \[\leadsto \color{blue}{x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot -27 + \left(x.re\_m \cdot x.im\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re\_m \cdot \frac{x.re\_m + -19683}{x.im}\right)\\ \end{array} \end{array} \]

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 2.15e+96)
    (* x.im (+ (* x.re_m -27.0) (* (* x.re_m x.im) -2.0)))
    (* x.im (+ -2.0 (* x.re_m (/ (+ x.re_m -19683.0) 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 <= 2.15e+96) {
		tmp = x_46_im * ((x_46_re_m * -27.0) + ((x_46_re_m * x_46_im) * -2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15d+96) then
        tmp = x_46im * ((x_46re_m * (-27.0d0)) + ((x_46re_m * x_46im) * (-2.0d0)))
    else
        tmp = x_46im * ((-2.0d0) + (x_46re_m * ((x_46re_m + (-19683.0d0)) / 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 <= 2.15e+96) {
		tmp = x_46_im * ((x_46_re_m * -27.0) + ((x_46_re_m * x_46_im) * -2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15e+96:
		tmp = x_46_im * ((x_46_re_m * -27.0) + ((x_46_re_m * x_46_im) * -2.0))
	else:
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15e+96)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * -27.0) + Float64(Float64(x_46_re_m * x_46_im) * -2.0)));
	else
		tmp = Float64(x_46_im * Float64(-2.0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m + -19683.0) / 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 <= 2.15e+96)
		tmp = x_46_im * ((x_46_re_m * -27.0) + ((x_46_re_m * x_46_im) * -2.0));
	else
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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, 2.15e+96], N[(x$46$im * N[(N[(x$46$re$95$m * -27.0), $MachinePrecision] + N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-2.0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + -19683.0), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.15000000000000001e96

   1. Initial program 85.5%

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

      1. difference-of-squares 87.4%

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

   4. Applied egg-rr 87.4%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in x.re around 0 30.2%

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

      1. associate-*r* 30.2%

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

      2. *-commutative 30.2%

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

   9. Simplified 30.2%

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

   10. Taylor expanded in x.im around 0 33.0%

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

   if 2.15000000000000001e96 < x.re

   1. Initial program 72.5%

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

      1. difference-of-squares 75.0%

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

   4. Applied egg-rr 75.0%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in x.im around inf 44.9%

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

      1. unpow2 44.9%

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

      2. associate-*l* 44.9%

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

      3. associate-*r/ 47.5%

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

      4. distribute-lft-out 72.5%

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

      5. +-commutative 72.5%

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

      6. remove-double-neg 72.5%

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

      7. mul-1-neg 72.5%

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

      8. sub-neg 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-/l* 72.5%

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

      11. *-commutative 72.5%

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

      12. distribute-lft-out-- 72.5%

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

      13. sub-neg 72.5%

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

      14. metadata-eval 72.5%

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

      15. sub-neg 72.5%

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

      16. metadata-eval 72.5%

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

      17. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in x.im around 0 70.0%

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

   12. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{x.im \cdot \left({x.re}^{2} \cdot \frac{x.re + -27}{x.im}\right) + x.im \cdot \frac{0}{0}} \]

   14. Simplified 80.9%

       \[\leadsto \color{blue}{x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27 + \left(x.re \cdot x.im\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re\_m \cdot \frac{x.re\_m + -19683}{x.im}\right)\\ \end{array} \end{array} \]

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 2.15e+96)
    (* (* x.re_m x.im) (- -27.0 (* x.im 2.0)))
    (* x.im (+ -2.0 (* x.re_m (/ (+ x.re_m -19683.0) 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 <= 2.15e+96) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15d+96) then
        tmp = (x_46re_m * x_46im) * ((-27.0d0) - (x_46im * 2.0d0))
    else
        tmp = x_46im * ((-2.0d0) + (x_46re_m * ((x_46re_m + (-19683.0d0)) / 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 <= 2.15e+96) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15e+96:
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0))
	else:
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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 <= 2.15e+96)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(-27.0 - Float64(x_46_im * 2.0)));
	else
		tmp = Float64(x_46_im * Float64(-2.0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m + -19683.0) / 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 <= 2.15e+96)
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	else
		tmp = x_46_im * (-2.0 + (x_46_re_m * ((x_46_re_m + -19683.0) / 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, 2.15e+96], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(-27.0 - N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-2.0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m + -19683.0), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.15000000000000001e96

   1. Initial program 85.5%

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

      1. difference-of-squares 87.4%

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

   4. Applied egg-rr 87.4%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in x.re around 0 30.2%

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

      1. associate-*r* 30.2%

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

      2. *-commutative 30.2%

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

   9. Simplified 30.2%

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

       1. *-commutative 85.5%

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

       2. *-un-lft-identity 85.5%

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

       3. *-un-lft-identity 85.5%

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

       4. distribute-rgt-out 85.5%

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

       5. metadata-eval 85.5%

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

   11. Applied egg-rr 30.2%

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

       1. associate-*l* 30.7%

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

       2. *-commutative 30.7%

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

       3. associate-*r* 30.2%

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

       4. associate-*l* 30.2%

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

       5. *-commutative 30.2%

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

       6. distribute-lft-out-- 32.5%

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

   13. Applied egg-rr 32.5%

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

   if 2.15000000000000001e96 < x.re

   1. Initial program 72.5%

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

      1. difference-of-squares 75.0%

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

   4. Applied egg-rr 75.0%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in x.im around inf 44.9%

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

      1. unpow2 44.9%

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

      2. associate-*l* 44.9%

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

      3. associate-*r/ 47.5%

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

      4. distribute-lft-out 72.5%

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

      5. +-commutative 72.5%

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

      6. remove-double-neg 72.5%

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

      7. mul-1-neg 72.5%

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

      8. sub-neg 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-/l* 72.5%

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

      11. *-commutative 72.5%

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

      12. distribute-lft-out-- 72.5%

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

      13. sub-neg 72.5%

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

      14. metadata-eval 72.5%

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

      15. sub-neg 72.5%

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

      16. metadata-eval 72.5%

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

      17. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in x.im around 0 70.0%

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

   12. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{x.im \cdot \left({x.re}^{2} \cdot \frac{x.re + -27}{x.im}\right) + x.im \cdot \frac{0}{0}} \]

   14. Simplified 80.9%

       \[\leadsto \color{blue}{x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-2 + x.re \cdot \frac{x.re + -19683}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \frac{x.re\_m + -19683}{x.im} - x.im\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.15e+96)
    (* (* x.re_m x.im) (- -27.0 (* x.im 2.0)))
    (- (* x.re_m (/ (+ x.re_m -19683.0) x.im)) x.im))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.15e+96) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.15d+96) then
        tmp = (x_46re_m * x_46im) * ((-27.0d0) - (x_46im * 2.0d0))
    else
        tmp = (x_46re_m * ((x_46re_m + (-19683.0d0)) / x_46im)) - x_46im
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.15e+96) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)) - x_46_im;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.15e+96:
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0))
	else:
		tmp = (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)) - x_46_im
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.15e+96)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(-27.0 - Float64(x_46_im * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m + -19683.0) / x_46_im)) - x_46_im);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.15e+96)
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	else
		tmp = (x_46_re_m * ((x_46_re_m + -19683.0) / x_46_im)) - x_46_im;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.15e+96], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(-27.0 - N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m + -19683.0), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.15000000000000001e96

   1. Initial program 85.5%

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

      1. difference-of-squares 87.4%

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

   4. Applied egg-rr 87.4%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in x.re around 0 30.2%

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

      1. associate-*r* 30.2%

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

      2. *-commutative 30.2%

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

   9. Simplified 30.2%

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

       1. *-commutative 85.5%

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

       2. *-un-lft-identity 85.5%

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

       3. *-un-lft-identity 85.5%

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

       4. distribute-rgt-out 85.5%

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

       5. metadata-eval 85.5%

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

   11. Applied egg-rr 30.2%

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

       1. associate-*l* 30.7%

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

       2. *-commutative 30.7%

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

       3. associate-*r* 30.2%

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

       4. associate-*l* 30.2%

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

       5. *-commutative 30.2%

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

       6. distribute-lft-out-- 32.5%

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

   13. Applied egg-rr 32.5%

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

   if 2.15000000000000001e96 < x.re

   1. Initial program 72.5%

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

      1. difference-of-squares 75.0%

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

   4. Applied egg-rr 75.0%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in x.im around inf 44.9%

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

      1. unpow2 44.9%

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

      2. associate-*l* 44.9%

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

      3. associate-*r/ 47.5%

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

      4. distribute-lft-out 72.5%

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

      5. +-commutative 72.5%

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

      6. remove-double-neg 72.5%

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

      7. mul-1-neg 72.5%

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

      8. sub-neg 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-/l* 72.5%

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

      11. *-commutative 72.5%

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

      12. distribute-lft-out-- 72.5%

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

      13. sub-neg 72.5%

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

      14. metadata-eval 72.5%

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

      15. sub-neg 72.5%

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

      16. metadata-eval 72.5%

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

      17. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in x.im around 0 70.0%

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

       1. associate-*r* 70.0%

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

       2. fma-neg 70.0%

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

       3. associate-/l* 70.0%

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

       4. sub-neg 70.0%

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

       5. metadata-eval 70.0%

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

       6. *-commutative 70.0%

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

       7. distribute-rgt-neg-in 70.0%

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

       8. *-commutative 70.0%

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

       9. flip-+ 0.0%

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

       10. +-inverses 0.0%

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

       11. metadata-eval 0.0%

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

       12. +-inverses 0.0%

           \[\leadsto \mathsf{fma}\left(x.im \cdot x.re, x.re \cdot \frac{x.re + -27}{x.im}, x.im \cdot \left(-\frac{\log 1}{\color{blue}{0}}\right)\right) \]

       13. metadata-eval 0.0%

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

       14. distribute-neg-frac2 0.0%

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

       15. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(x.im \cdot x.re, x.re \cdot \frac{x.re + -27}{x.im}, x.im \cdot \frac{\color{blue}{0}}{-\log 1}\right) \]

       16. metadata-eval 0.0%

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

       17. metadata-eval 0.0%

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

   12. Applied egg-rr 0.0%

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

   14. Simplified 33.3%

       \[\leadsto \color{blue}{x.re \cdot \frac{x.re + -19683}{x.im} + \left(-x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \frac{x.re + -19683}{x.im} - x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.45 \cdot 10^{+157}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(-27 + x.im \cdot 2\right)\\ \end{array} \end{array} \]

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 2.45e+157)
    (* (* x.re_m x.im) (- -27.0 (* x.im 2.0)))
    (* (* x.re_m x.im) (+ -27.0 (* x.im 2.0))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.45e+157) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.45d+157) then
        tmp = (x_46re_m * x_46im) * ((-27.0d0) - (x_46im * 2.0d0))
    else
        tmp = (x_46re_m * x_46im) * ((-27.0d0) + (x_46im * 2.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.45e+157) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	} else {
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.45e+157:
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0))
	else:
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.45e+157)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(-27.0 - Float64(x_46_im * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(-27.0 + Float64(x_46_im * 2.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.45e+157)
		tmp = (x_46_re_m * x_46_im) * (-27.0 - (x_46_im * 2.0));
	else
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.45e+157], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(-27.0 - N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(-27.0 + N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.4500000000000001e157

   1. Initial program 86.2%

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

      1. difference-of-squares 88.0%

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

   4. Applied egg-rr 88.0%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in x.re around 0 28.9%

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

      1. associate-*r* 28.9%

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

      2. *-commutative 28.9%

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

   9. Simplified 28.9%

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

       1. *-commutative 86.2%

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

       2. *-un-lft-identity 86.2%

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

       3. *-un-lft-identity 86.2%

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

       4. distribute-rgt-out 86.2%

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

       5. metadata-eval 86.2%

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

   11. Applied egg-rr 28.9%

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

       1. associate-*l* 29.3%

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

       2. *-commutative 29.3%

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

       3. associate-*r* 28.9%

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

       4. associate-*l* 28.9%

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

       5. *-commutative 28.9%

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

       6. distribute-lft-out-- 31.5%

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

   13. Applied egg-rr 31.5%

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

   if 2.4500000000000001e157 < x.re

   1. Initial program 62.1%

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

      1. difference-of-squares 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in x.re around 0 4.9%

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

      1. associate-*r* 4.9%

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

      2. *-commutative 4.9%

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

   9. Simplified 4.9%

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

       1. *-commutative 62.1%

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

       2. *-un-lft-identity 62.1%

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

       3. *-un-lft-identity 62.1%

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

       4. distribute-rgt-out 62.1%

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

       5. metadata-eval 62.1%

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

   11. Applied egg-rr 4.9%

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

   13. Applied egg-rr 36.2%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.45 \cdot 10^{+157}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-27 - x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-27 + x.im \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(-27 + x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re\_m \cdot \left(x.im \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 6e+183)
    (* (* x.re_m x.im) (+ -27.0 (* x.im 2.0)))
    (+ x.im (* x.re_m (* x.im -27.0))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 6e+183) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	} else {
		tmp = x_46_im + (x_46_re_m * (x_46_im * -27.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 6d+183) then
        tmp = (x_46re_m * x_46im) * ((-27.0d0) + (x_46im * 2.0d0))
    else
        tmp = x_46im + (x_46re_m * (x_46im * (-27.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 6e+183) {
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	} else {
		tmp = x_46_im + (x_46_re_m * (x_46_im * -27.0));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 6e+183:
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0))
	else:
		tmp = x_46_im + (x_46_re_m * (x_46_im * -27.0))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6e+183)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(-27.0 + Float64(x_46_im * 2.0)));
	else
		tmp = Float64(x_46_im + Float64(x_46_re_m * Float64(x_46_im * -27.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 6e+183)
		tmp = (x_46_re_m * x_46_im) * (-27.0 + (x_46_im * 2.0));
	else
		tmp = x_46_im + (x_46_re_m * (x_46_im * -27.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 6e+183], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(-27.0 + N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im + N[(x$46$re$95$m * N[(x$46$im * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.99999999999999992e183

   1. Initial program 84.9%

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

      1. difference-of-squares 85.8%

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

   4. Applied egg-rr 85.8%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in x.re around 0 21.2%

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

      1. associate-*r* 21.2%

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

      2. *-commutative 21.2%

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

   9. Simplified 21.2%

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

       1. *-commutative 84.9%

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

       2. *-un-lft-identity 84.9%

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

       3. *-un-lft-identity 84.9%

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

       4. distribute-rgt-out 84.9%

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

       5. metadata-eval 84.9%

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

   11. Applied egg-rr 21.2%

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

   13. Applied egg-rr 18.0%

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

   if 5.99999999999999992e183 < x.im

   1. Initial program 66.2%

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

      1. difference-of-squares 81.2%

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

   4. Applied egg-rr 81.2%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in x.re around 0 84.0%

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

      1. associate-*r* 84.0%

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

      2. *-commutative 84.0%

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

   9. Simplified 84.0%

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

       1. *-commutative 66.2%

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

       2. *-un-lft-identity 66.2%

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

       3. *-un-lft-identity 66.2%

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

       4. distribute-rgt-out 66.2%

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

       5. metadata-eval 66.2%

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

   11. Applied egg-rr 84.0%

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

       1. add-log-exp 81.8%

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

       2. *-commutative 81.8%

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

       3. exp-lft-sqr 81.8%

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

       4. exp-sum 81.8%

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

       5. add-log-exp 84.0%

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

       6. *-commutative 84.0%

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

   13. Applied egg-rr 0.0%

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

   15. Simplified 23.3%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-27 + x.im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im + x.re \cdot \left(x.im \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 85.4%

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

4. Applied egg-rr 85.4%

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

6. Simplified 51.8%

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

7. Taylor expanded in x.re around 0 26.1%

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

   1. associate-*r* 26.1%

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

   2. *-commutative 26.1%

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

9. Simplified 26.1%

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

    1. *-commutative 83.5%

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

    2. *-un-lft-identity 83.5%

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

    3. *-un-lft-identity 83.5%

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

    4. distribute-rgt-out 83.5%

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

    5. metadata-eval 83.5%

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

11. Applied egg-rr 26.1%

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

    1. add-log-exp 22.0%

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

    2. *-commutative 22.0%

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

    3. exp-lft-sqr 22.0%

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

    4. exp-sum 22.0%

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

    5. add-log-exp 26.1%

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

    6. *-commutative 26.1%

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

13. Applied egg-rr 0.0%

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

15. Simplified 6.6%

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

16. Final simplification 6.6%

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

Alternative 11: 2.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(-19683 - x.im\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im) :precision binary64 (* x.re_s (- -19683.0 x.im)))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-19683.0 - x_46_im);
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((-19683.0d0) - x_46im)
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-19683.0 - x_46_im);
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (-19683.0 - x_46_im)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(-19683.0 - x_46_im))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (-19683.0 - x_46_im);
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(-19683.0 - x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 85.4%

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

4. Applied egg-rr 85.4%

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

6. Simplified 51.8%

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

7. Taylor expanded in x.re around 0 26.1%

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

   1. associate-*r* 26.1%

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

   2. *-commutative 26.1%

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

9. Simplified 26.1%

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

    1. *-commutative 83.5%

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

    2. *-un-lft-identity 83.5%

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

    3. *-un-lft-identity 83.5%

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

    4. distribute-rgt-out 83.5%

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

    5. metadata-eval 83.5%

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

11. Applied egg-rr 26.1%

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

12. Taylor expanded in x.im around 0 28.9%

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

14. Simplified 3.1%

    \[\leadsto \color{blue}{\left(-x.im\right) + -19683} \]

15. Final simplification 3.1%

    \[\leadsto -19683 - x.im \]
16. Add Preprocessing

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot -0.5 \end{array} \]

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 -0.5))

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 * -0.5;
}

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 * (-0.5d0)
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 * -0.5;
}

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 * -0.5

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 * -0.5)
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 * -0.5;
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 * -0.5), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Simplified 82.7%

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

   1. associate-*r* 82.7%

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

   2. associate-*l* 82.7%

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

   3. +-commutative 82.7%

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

   4. associate-*r* 89.7%

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

   5. associate-*r* 89.7%

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

   6. fma-define 92.0%

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

6. Applied egg-rr 92.0%

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

8. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-0.5} \]
9. Add Preprocessing

Alternative 13: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot -757 \end{array} \]

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 -757.0))

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 * -757.0;
}

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 * (-757.0d0)
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 * -757.0;
}

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 * -757.0

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 * -757.0)
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 * -757.0;
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 * -757.0), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Simplified 82.7%

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

   1. associate-*r* 82.7%

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

   2. associate-*l* 82.7%

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

   3. +-commutative 82.7%

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

   4. associate-*r* 89.7%

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

   5. associate-*r* 89.7%

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

   6. fma-define 92.0%

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

6. Applied egg-rr 92.0%

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

8. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-757} \]
9. Add Preprocessing

Developer Target 1: 87.3% 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 2024139 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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