math.cube on complex, real part

Percentage Accurate: 82.7% → 99.9%

Time: 7.6s

Alternatives: 9

Speedup: 0.5×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) \cdot -3 + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5.8e+102)
    (+ (* (* x.im_m (* x.re_m x.im_m)) -3.0) (pow x.re_m 3.0))
    (* x.re_m (* (- x.re_m x.im_m) (+ x.re_m -27.0))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 5.8e+102) {
		tmp = ((x_46_im_m * (x_46_re_m * x_46_im_m)) * -3.0) + pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 5.8d+102) then
        tmp = ((x_46im_m * (x_46re_m * x_46im_m)) * (-3.0d0)) + (x_46re_m ** 3.0d0)
    else
        tmp = x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + (-27.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 5.8e+102) {
		tmp = ((x_46_im_m * (x_46_re_m * x_46_im_m)) * -3.0) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 5.8e+102:
		tmp = ((x_46_im_m * (x_46_re_m * x_46_im_m)) * -3.0) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 5.8e+102)
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m)) * -3.0) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + -27.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 5.8e+102)
		tmp = ((x_46_im_m * (x_46_re_m * x_46_im_m)) * -3.0) + (x_46_re_m ^ 3.0);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5.8e+102], N[(N[(N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.8000000000000005e102

   1. Initial program 85.1%

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

   3. Simplified 84.3%

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

      1. associate-*r* 84.3%

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

      2. associate-*l* 84.3%

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

      3. +-commutative 84.3%

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

      4. associate-*r* 92.0%

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

      5. associate-*r* 92.0%

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

      6. fma-define 92.9%

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

   6. Applied egg-rr 92.9%

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

      1. fma-undefine 92.0%

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

      2. associate-*r* 92.0%

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

   8. Applied egg-rr 92.0%

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

   if 5.8000000000000005e102 < x.re

   1. Initial program 69.2%

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

      1. difference-of-squares 74.4%

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

   4. Applied egg-rr 74.4%

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

   6. Simplified 69.2%

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

   8. Applied egg-rr 92.3%

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

      1. +-rgt-identity 92.3%

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

      2. associate-*l* 92.3%

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

   10. Simplified 92.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot x.im\right)\right) \cdot -3 + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq 10^{-261}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right)\right) - x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot 0 - x.re\_m \cdot \left(\left(x.re\_m + x.im\_m\right) \cdot \left(x.im\_m - x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       1e-261)
    (-
     (* x.re_m (* (- x.re_m x.im_m) (+ x.re_m x.im_m)))
     (* x.im_m (* x.im_m (* x.re_m 2.0))))
    (- (* x.im_m 0.0) (* x.re_m (* (+ x.re_m x.im_m) (- x.im_m x.re_m)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-261) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_im_m * (x_46_re_m * 2.0)));
	} else {
		tmp = (x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= 1d-261) then
        tmp = (x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + x_46im_m))) - (x_46im_m * (x_46im_m * (x_46re_m * 2.0d0)))
    else
        tmp = (x_46im_m * 0.0d0) - (x_46re_m * ((x_46re_m + x_46im_m) * (x_46im_m - x_46re_m)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-261) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_im_m * (x_46_re_m * 2.0)));
	} else {
		tmp = (x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-261:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_im_m * (x_46_re_m * 2.0)))
	else:
		tmp = (x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= 1e-261)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m))) - Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m * 2.0))));
	else
		tmp = Float64(Float64(x_46_im_m * 0.0) - Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_im_m - x_46_re_m))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-261)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * (x_46_im_m * (x_46_re_m * 2.0)));
	else
		tmp = (x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-261], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * 0.0), $MachinePrecision] - N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999984e-262

   1. Initial program 93.9%

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

      1. difference-of-squares 93.9%

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in x.re around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 93.9%

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

      3. *-commutative 93.9%

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

   7. Simplified 93.9%

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

   if 9.99999999999999984e-262 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 69.0%

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

      1. difference-of-squares 72.5%

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

   4. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. add-sqr-sqrt 38.2%

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

      3. sqrt-unprod 38.6%

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

      4. sqr-neg 38.6%

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

      5. mul-1-neg 38.6%

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

      6. mul-1-neg 38.6%

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

      7. sqrt-unprod 14.2%

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

      8. add-sqr-sqrt 55.0%

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

      9. mul-1-neg 55.0%

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

      10. cancel-sign-sub-inv 55.0%

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

      11. +-inverses 77.6%

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

   6. Applied egg-rr 77.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-261}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot 0 - x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.im - x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -2 + x.re\_m \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.6e+18)
    (* x.im_m (+ (* (* x.re_m x.im_m) -2.0) (* x.re_m -27.0)))
    (* x.re_m (* (- x.re_m x.im_m) (+ x.re_m -27.0))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.6e+18) {
		tmp = x_46_im_m * (((x_46_re_m * x_46_im_m) * -2.0) + (x_46_re_m * -27.0));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 2.6d+18) then
        tmp = x_46im_m * (((x_46re_m * x_46im_m) * (-2.0d0)) + (x_46re_m * (-27.0d0)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + (-27.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 2.6e+18) {
		tmp = x_46_im_m * (((x_46_re_m * x_46_im_m) * -2.0) + (x_46_re_m * -27.0));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 2.6e+18:
		tmp = x_46_im_m * (((x_46_re_m * x_46_im_m) * -2.0) + (x_46_re_m * -27.0))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 2.6e+18)
		tmp = Float64(x_46_im_m * Float64(Float64(Float64(x_46_re_m * x_46_im_m) * -2.0) + Float64(x_46_re_m * -27.0)));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + -27.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 2.6e+18)
		tmp = x_46_im_m * (((x_46_re_m * x_46_im_m) * -2.0) + (x_46_re_m * -27.0));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.6e+18], N[(x$46$im$95$m * N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -2.0), $MachinePrecision] + N[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.6e18

   1. Initial program 84.2%

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

      1. difference-of-squares 85.2%

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

   4. Applied egg-rr 85.2%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in x.re around 0 31.0%

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

      1. *-commutative 31.0%

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

   9. Simplified 31.0%

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

   10. Taylor expanded in x.im around 0 32.5%

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

   if 2.6e18 < x.re

   1. Initial program 76.9%

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

      1. difference-of-squares 80.7%

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

   4. Applied egg-rr 80.7%

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

   6. Simplified 71.1%

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

   8. Applied egg-rr 88.4%

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

      1. +-rgt-identity 88.4%

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

      2. associate-*l* 88.4%

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

   10. Simplified 88.4%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2 + x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 27.0) {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 27.0d0) then
        tmp = x_46re_m * (x_46re_m * (-27.0d0))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im_m) * (x_46re_m + (-27.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 27.0) {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 27.0:
		tmp = x_46_re_m * (x_46_re_m * -27.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 27.0)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * -27.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + -27.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 27.0)
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + -27.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x.re < 27

   1. Initial program 83.9%

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

      1. difference-of-squares 84.9%

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

   4. Applied egg-rr 84.9%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in x.im around 0 31.3%

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

      1. *-commutative 31.3%

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

      2. sub-neg 31.3%

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

      3. metadata-eval 31.3%

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

      4. associate-*l* 32.8%

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

      5. *-commutative 32.8%

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

      6. unpow2 32.8%

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

      7. sub-neg 32.8%

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

      8. metadata-eval 32.8%

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

      9. associate-*l* 32.8%

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

      10. metadata-eval 32.8%

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

      11. sub-neg 32.8%

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

      12. distribute-lft-in 39.8%

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

      13. sub-neg 39.8%

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

      14. metadata-eval 39.8%

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

      15. distribute-rgt-out 41.3%

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

   9. Simplified 41.3%

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

   10. Taylor expanded in x.im around 0 42.9%

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

   11. Taylor expanded in x.re around 0 34.6%

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

       1. *-commutative 34.6%

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

   13. Simplified 34.6%

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

   if 27 < x.re

   1. Initial program 78.4%

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

      1. difference-of-squares 82.0%

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

   4. Applied egg-rr 82.0%

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

   6. Simplified 70.0%

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

   8. Applied egg-rr 83.8%

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

      1. +-rgt-identity 83.8%

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

      2. associate-*l* 83.8%

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

   10. Simplified 83.8%

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

4. Final simplification 45.4%

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

Alternative 5: 78.5% accurate, 1.5× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (- (* x.im_m 0.0) (* x.re_m (* (+ x.re_m x.im_m) (- x.im_m x.re_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * ((x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m))));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((x_46im_m * 0.0d0) - (x_46re_m * ((x_46re_m + x_46im_m) * (x_46im_m - x_46re_m))))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * ((x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m))));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * ((x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m))))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(Float64(x_46_im_m * 0.0) - Float64(x_46_re_m * Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_im_m - x_46_re_m)))))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * ((x_46_im_m * 0.0) - (x_46_re_m * ((x_46_re_m + x_46_im_m) * (x_46_im_m - x_46_re_m))));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * N[(N[(x$46$im$95$m * 0.0), $MachinePrecision] - N[(x$46$re$95$m * N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. difference-of-squares 84.3%

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

4. Applied egg-rr 84.3%

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

   1. *-commutative 84.3%

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

   2. add-sqr-sqrt 40.4%

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

   3. sqrt-unprod 67.0%

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

   4. sqr-neg 67.0%

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

   5. mul-1-neg 67.0%

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

   6. mul-1-neg 67.0%

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

   7. sqrt-unprod 34.3%

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

   8. add-sqr-sqrt 67.3%

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

   9. mul-1-neg 67.3%

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

   10. cancel-sign-sub-inv 67.3%

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

   11. +-inverses 78.7%

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

6. Applied egg-rr 78.7%

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

7. Final simplification 78.7%

   \[\leadsto x.im \cdot 0 - x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.im - x.re\right)\right) \]
8. Add Preprocessing

Alternative 6: 40.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.3 \cdot 10^{+238}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 3.3e+238)
    (* x.im_m (* x.re_m (- x.re_m 27.0)))
    (* x.im_m (* x.re_m -27.0)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.3e+238) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m - 27.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 3.3d+238) then
        tmp = x_46im_m * (x_46re_m * (x_46re_m - 27.0d0))
    else
        tmp = x_46im_m * (x_46re_m * (-27.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.3e+238) {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m - 27.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 3.3e+238:
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m - 27.0))
	else:
		tmp = x_46_im_m * (x_46_re_m * -27.0)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.3e+238)
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m - 27.0)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * -27.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 3.3e+238)
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m - 27.0));
	else
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 3.3e+238], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.3000000000000001e238

   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 84.7%

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

   4. Applied egg-rr 84.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in x.im around 0 37.6%

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

      1. *-commutative 37.6%

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

      2. sub-neg 37.6%

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

      3. metadata-eval 37.6%

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

      4. associate-*l* 41.7%

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

      5. *-commutative 41.7%

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

      6. unpow2 41.7%

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

      7. sub-neg 41.7%

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

      8. metadata-eval 41.7%

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

      9. associate-*l* 41.7%

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

      10. metadata-eval 41.7%

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

      11. sub-neg 41.7%

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

      12. distribute-lft-in 49.8%

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

      13. sub-neg 49.8%

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

      14. metadata-eval 49.8%

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

      15. distribute-rgt-out 52.3%

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

   9. Simplified 52.3%

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

   10. Taylor expanded in x.im around inf 28.1%

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

   if 3.3000000000000001e238 < x.im

   1. Initial program 65.9%

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

      1. difference-of-squares 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 51.0%

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

   7. Taylor expanded in x.im around 0 12.2%

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

      1. *-commutative 12.2%

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

      2. sub-neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. associate-*l* 12.2%

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

      5. *-commutative 12.2%

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

      6. unpow2 12.2%

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

      7. sub-neg 12.2%

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

      8. metadata-eval 12.2%

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

      9. associate-*l* 12.2%

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

      10. metadata-eval 12.2%

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

      11. sub-neg 12.2%

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

      12. distribute-lft-in 12.2%

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

      13. sub-neg 12.2%

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

      14. metadata-eval 12.2%

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

      15. distribute-rgt-out 21.3%

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

   9. Simplified 21.3%

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

   10. Taylor expanded in x.re around 0 40.0%

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

       1. pow1 40.0%

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

       2. *-commutative 40.0%

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

       3. associate-*l* 40.0%

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

   12. Applied egg-rr 40.0%

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

       1. unpow1 40.0%

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

   14. Simplified 40.0%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.3 \cdot 10^{+238}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.4 \cdot 10^{+238}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 2.4e+238)
    (* x.re_m (* x.re_m (- x.re_m 27.0)))
    (* x.im_m (* x.re_m -27.0)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.4e+238) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m - 27.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2.4d+238) then
        tmp = x_46re_m * (x_46re_m * (x_46re_m - 27.0d0))
    else
        tmp = x_46im_m * (x_46re_m * (-27.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.4e+238) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m - 27.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2.4e+238:
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m - 27.0))
	else:
		tmp = x_46_im_m * (x_46_re_m * -27.0)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.4e+238)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_re_m - 27.0)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * -27.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2.4e+238)
		tmp = x_46_re_m * (x_46_re_m * (x_46_re_m - 27.0));
	else
		tmp = x_46_im_m * (x_46_re_m * -27.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2.4e+238], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.4e238

   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 84.7%

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

   4. Applied egg-rr 84.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in x.im around 0 37.6%

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

      1. *-commutative 37.6%

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

      2. sub-neg 37.6%

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

      3. metadata-eval 37.6%

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

      4. associate-*l* 41.7%

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

      5. *-commutative 41.7%

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

      6. unpow2 41.7%

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

      7. sub-neg 41.7%

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

      8. metadata-eval 41.7%

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

      9. associate-*l* 41.7%

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

      10. metadata-eval 41.7%

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

      11. sub-neg 41.7%

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

      12. distribute-lft-in 49.8%

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

      13. sub-neg 49.8%

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

      14. metadata-eval 49.8%

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

      15. distribute-rgt-out 52.3%

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

   9. Simplified 52.3%

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

   10. Taylor expanded in x.im around 0 52.1%

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

   if 2.4e238 < x.im

   1. Initial program 65.9%

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

      1. difference-of-squares 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 51.0%

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

   7. Taylor expanded in x.im around 0 12.2%

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

      1. *-commutative 12.2%

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

      2. sub-neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. associate-*l* 12.2%

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

      5. *-commutative 12.2%

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

      6. unpow2 12.2%

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

      7. sub-neg 12.2%

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

      8. metadata-eval 12.2%

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

      9. associate-*l* 12.2%

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

      10. metadata-eval 12.2%

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

      11. sub-neg 12.2%

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

      12. distribute-lft-in 12.2%

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

      13. sub-neg 12.2%

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

      14. metadata-eval 12.2%

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

      15. distribute-rgt-out 21.3%

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

   9. Simplified 21.3%

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

   10. Taylor expanded in x.re around 0 40.0%

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

       1. pow1 40.0%

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

       2. *-commutative 40.0%

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

       3. associate-*l* 40.0%

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

   12. Applied egg-rr 40.0%

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

       1. unpow1 40.0%

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

   14. Simplified 40.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.4 \cdot 10^{+238}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 21.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 9.5e-167)
    (* x.re_m (* x.re_m -27.0))
    (* -27.0 (* x.re_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.5e-167) {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	} else {
		tmp = -27.0 * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 9.5d-167) then
        tmp = x_46re_m * (x_46re_m * (-27.0d0))
    else
        tmp = (-27.0d0) * (x_46re_m * x_46im_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 9.5e-167) {
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	} else {
		tmp = -27.0 * (x_46_re_m * x_46_im_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 9.5e-167:
		tmp = x_46_re_m * (x_46_re_m * -27.0)
	else:
		tmp = -27.0 * (x_46_re_m * x_46_im_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 9.5e-167)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * -27.0));
	else
		tmp = Float64(-27.0 * Float64(x_46_re_m * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 9.5e-167)
		tmp = x_46_re_m * (x_46_re_m * -27.0);
	else
		tmp = -27.0 * (x_46_re_m * x_46_im_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 9.5e-167], N[(x$46$re$95$m * N[(x$46$re$95$m * -27.0), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.49999999999999955e-167

   1. Initial program 82.5%

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

      1. difference-of-squares 83.8%

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

   4. Applied egg-rr 83.8%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in x.im around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. sub-neg 38.4%

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

      3. metadata-eval 38.4%

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

      4. associate-*l* 40.3%

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

      5. *-commutative 40.3%

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

      6. unpow2 40.3%

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

      7. sub-neg 40.3%

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

      8. metadata-eval 40.3%

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

      9. associate-*l* 40.3%

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

      10. metadata-eval 40.3%

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

      11. sub-neg 40.3%

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

      12. distribute-lft-in 49.2%

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

      13. sub-neg 49.2%

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

      14. metadata-eval 49.2%

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

      15. distribute-rgt-out 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in x.im around 0 53.1%

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

   11. Taylor expanded in x.re around 0 42.5%

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

       1. *-commutative 42.5%

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

   13. Simplified 42.5%

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

   if 9.49999999999999955e-167 < x.re

   1. Initial program 83.0%

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

      1. difference-of-squares 85.0%

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

   4. Applied egg-rr 85.0%

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

   6. Simplified 51.7%

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

   7. Taylor expanded in x.im around 0 33.5%

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

      1. *-commutative 33.5%

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

      2. sub-neg 33.5%

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

      3. metadata-eval 33.5%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. unpow2 40.6%

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

      7. sub-neg 40.6%

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

      8. metadata-eval 40.6%

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

      9. associate-*l* 40.6%

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

      10. metadata-eval 40.6%

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

      11. sub-neg 40.6%

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

      12. distribute-lft-in 46.6%

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

      13. sub-neg 46.6%

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

      14. metadata-eval 46.6%

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

      15. distribute-rgt-out 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in x.re around 0 9.1%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 20.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (-27.0 * (x_46_re_m * x_46_im_m));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((-27.0d0) * (x_46re_m * x_46im_m))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (-27.0 * (x_46_re_m * x_46_im_m));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (-27.0 * (x_46_re_m * x_46_im_m))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(-27.0 * Float64(x_46_re_m * x_46_im_m)))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (-27.0 * (x_46_re_m * x_46_im_m));
end

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. difference-of-squares 84.3%

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

4. Applied egg-rr 84.3%

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

6. Simplified 52.6%

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

7. Taylor expanded in x.im around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. sub-neg 36.5%

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

   3. metadata-eval 36.5%

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

   4. associate-*l* 40.4%

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

   5. *-commutative 40.4%

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

   6. unpow2 40.4%

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

   7. sub-neg 40.4%

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

   8. metadata-eval 40.4%

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

   9. associate-*l* 40.4%

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

   10. metadata-eval 40.4%

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

   11. sub-neg 40.4%

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

   12. distribute-lft-in 48.2%

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

   13. sub-neg 48.2%

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

   14. metadata-eval 48.2%

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

   15. distribute-rgt-out 51.0%

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

9. Simplified 51.0%

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

10. Taylor expanded in x.re around 0 19.0%

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

11. Final simplification 19.0%

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

Developer target: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024078 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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