math.cube on complex, real part

Percentage Accurate: 82.5% → 99.9%

Time: 7.2s

Alternatives: 11

Speedup: 1.1×

• 45.2% 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: 82.5% 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, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left({x.re\_m}^{3} - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot \left(x.re\_m - x.re\_m\right)\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+102)
    (-
     (-
      (pow x.re_m 3.0)
      (* x.im (+ (* x.re_m x.im) (* x.re_m (- x.re_m x.re_m)))))
     (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
    (+ (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+102) {
		tmp = (pow(x_46_re_m, 3.0) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * (x_46_re_m - x_46_re_m))))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2d+102) then
        tmp = ((x_46re_m ** 3.0d0) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * (x_46re_m - x_46re_m))))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+102) {
		tmp = (Math.pow(x_46_re_m, 3.0) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * (x_46_re_m - x_46_re_m))))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2e+102:
		tmp = (math.pow(x_46_re_m, 3.0) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * (x_46_re_m - x_46_re_m))))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2e+102)
		tmp = Float64(Float64((x_46_re_m ^ 3.0) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m))))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2e+102)
		tmp = ((x_46_re_m ^ 3.0) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * (x_46_re_m - x_46_re_m))))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+102], N[(N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.99999999999999995e102

   1. Initial program 88.7%

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

      1. difference-of-squares 90.0%

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

      2. *-commutative 90.0%

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

      3. +-commutative 90.0%

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

   4. Applied egg-rr 90.0%

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

   5. Taylor expanded in x.im around 0 87.7%

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

   if 1.99999999999999995e102 < x.re

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-neg-frac 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 71.4%

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

      1. difference-of-squares 74.3%

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

      2. *-commutative 74.3%

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

      3. +-commutative 74.3%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left({x.re}^{3} - x.im \cdot \left(x.re \cdot x.im + x.re \cdot \left(x.re - x.re\right)\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4 \cdot 10^{+102}:\\ \;\;\;\;{x.re\_m}^{3} + \left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4e+102)
    (+ (pow x.re_m 3.0) (* (* x.re_m x.im) (* x.im -3.0)))
    (+ (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4e+102) {
		tmp = pow(x_46_re_m, 3.0) + ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 4d+102) then
        tmp = (x_46re_m ** 3.0d0) + ((x_46re_m * x_46im) * (x_46im * (-3.0d0)))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4e+102) {
		tmp = Math.pow(x_46_re_m, 3.0) + ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 4e+102:
		tmp = math.pow(x_46_re_m, 3.0) + ((x_46_re_m * x_46_im) * (x_46_im * -3.0))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 4e+102)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 4e+102)
		tmp = (x_46_re_m ^ 3.0) + ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4e+102], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.99999999999999991e102

   1. Initial program 88.7%

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

   3. Simplified 85.1%

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

      1. associate-*r* 85.1%

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

      2. associate-*l* 85.1%

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

      3. +-commutative 85.1%

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

      4. associate-*r* 87.7%

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

      5. associate-*r* 87.6%

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

      6. fma-define 89.8%

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

      7. *-commutative 89.8%

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

   6. Applied egg-rr 89.8%

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

   8. Applied egg-rr 87.6%

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

   if 3.99999999999999991e102 < x.re

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-neg-frac 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 71.4%

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

      1. difference-of-squares 74.3%

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

      2. *-commutative 74.3%

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

      3. +-commutative 74.3%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+102}:\\ \;\;\;\;{x.re}^{3} + \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+102}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e+102)
    (-
     (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (+ (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+102) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2d+102) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2e+102) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2e+102:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2e+102)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2e+102)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+102], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.99999999999999995e102

   1. Initial program 88.7%

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

   3. Taylor expanded in x.re around 0 88.7%

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

   if 1.99999999999999995e102 < x.re

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-neg-frac 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 71.4%

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

      1. difference-of-squares 74.3%

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

      2. *-commutative 74.3%

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

      3. +-commutative 74.3%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot \left(\frac{x.im}{x.re\_m} + 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4.4e-110)
    (* -3.0 (* x.re_m (* x.im x.im)))
    (* (- x.re_m x.im) (* x.re_m (* x.re_m (+ (/ x.im x.re_m) 1.0)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.4e-110) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m * ((x_46_im / x_46_re_m) + 1.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 4.4d-110) then
        tmp = (-3.0d0) * (x_46re_m * (x_46im * x_46im))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m * ((x_46im / x_46re_m) + 1.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.4e-110) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m * ((x_46_im / x_46_re_m) + 1.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 4.4e-110:
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m * ((x_46_im / x_46_re_m) + 1.0)))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 4.4e-110)
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m * Float64(Float64(x_46_im / x_46_re_m) + 1.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 4.4e-110)
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m * ((x_46_im / x_46_re_m) + 1.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4.4e-110], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m * N[(N[(x$46$im / x$46$re$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.3999999999999999e-110

   1. Initial program 86.1%

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

   3. Taylor expanded in x.im around inf 53.6%

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

   4. Taylor expanded in x.re around 0 53.6%

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

   6. Applied egg-rr 53.6%

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

   if 4.3999999999999999e-110 < x.re

   1. Initial program 80.9%

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

      1. difference-of-squares 88.0%

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

      2. *-commutative 88.0%

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

      3. +-commutative 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in x.re around inf 88.0%

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

      1. sub-neg 88.0%

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

      2. associate-*l* 89.1%

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

      3. add-log-exp 61.2%

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

      4. neg-log 61.2%

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

      5. *-commutative 61.2%

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

      6. exp-prod 65.6%

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

      7. *-commutative 65.6%

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

      8. add-sqr-sqrt 65.6%

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

      9. sqrt-unprod 65.6%

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

      10. sqr-neg 65.6%

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

      11. mul-1-neg 65.6%

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

      12. mul-1-neg 65.6%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 62.3%

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

      15. mul-1-neg 62.3%

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

      16. cancel-sign-sub-inv 62.3%

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

      17. +-inverses 82.6%

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

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re \cdot \left(1 + \frac{x.im}{x.re}\right)\right) \cdot x.re\right) + 0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re \cdot \left(\frac{x.im}{x.re} + 1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 9 \cdot 10^{-89}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 9e-89)
    (* -3.0 (* x.re_m (* x.im x.im)))
    (+ (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 9e-89) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 9d-89) then
        tmp = (-3.0d0) * (x_46re_m * (x_46im * x_46im))
    else
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 9e-89) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 9e-89:
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im))
	else:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 9e-89)
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 9e-89)
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	else
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 9e-89], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.9999999999999998e-89

   1. Initial program 85.9%

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

   3. Taylor expanded in x.im around inf 53.6%

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

   4. Taylor expanded in x.re around 0 53.6%

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

   6. Applied egg-rr 53.6%

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

   if 8.9999999999999998e-89 < x.re

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-neg-frac 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 71.9%

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

      1. difference-of-squares 88.5%

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

      2. *-commutative 88.5%

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

      3. +-commutative 88.5%

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

   8. Applied egg-rr 84.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 9 \cdot 10^{-89}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 6 \cdot 10^{-88}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 6e-88)
    (* -3.0 (* x.re_m (* x.im x.im)))
    (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 6e-88) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 6d-88) then
        tmp = (-3.0d0) * (x_46re_m * (x_46im * x_46im))
    else
        tmp = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 6e-88) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 6e-88:
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 6e-88)
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 6e-88)
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 6e-88], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.9999999999999999e-88

   1. Initial program 85.9%

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

   3. Taylor expanded in x.im around inf 53.6%

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

   4. Taylor expanded in x.re around 0 53.6%

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

   6. Applied egg-rr 53.6%

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

   if 5.9999999999999999e-88 < x.re

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-neg-frac 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in x.im around 0 71.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6 \cdot 10^{-88}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.5 \cdot 10^{+211}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.re\_m \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4.5e+211)
    (* -3.0 (* x.re_m (* x.im x.im)))
    (* (* x.im x.im) (* x.re_m 3.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.5e+211) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_im * x_46_im) * (x_46_re_m * 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 4.5d+211) then
        tmp = (-3.0d0) * (x_46re_m * (x_46im * x_46im))
    else
        tmp = (x_46im * x_46im) * (x_46re_m * 3.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.5e+211) {
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	} else {
		tmp = (x_46_im * x_46_im) * (x_46_re_m * 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 4.5e+211:
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im))
	else:
		tmp = (x_46_im * x_46_im) * (x_46_re_m * 3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 4.5e+211)
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re_m * 3.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 4.5e+211)
		tmp = -3.0 * (x_46_re_m * (x_46_im * x_46_im));
	else
		tmp = (x_46_im * x_46_im) * (x_46_re_m * 3.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4.5e+211], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.5e211

   1. Initial program 86.3%

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

   3. Taylor expanded in x.im around inf 52.0%

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

   4. Taylor expanded in x.re around 0 52.0%

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

   6. Applied egg-rr 52.0%

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

   if 4.5e211 < x.re

   1. Initial program 53.3%

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

   3. Taylor expanded in x.im around inf 0.6%

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

   5. Applied egg-rr 0.6%

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

   7. Applied egg-rr 48.1%

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

      1. mul-1-neg 48.1%

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

      2. distribute-rgt-neg-in 48.1%

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

      3. metadata-eval 48.1%

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

   9. Simplified 48.1%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.5 \cdot 10^{+211}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Taylor expanded in x.im around inf 48.9%

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

4. Taylor expanded in x.re around 0 49.0%

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

6. Applied egg-rr 49.0%

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

7. Final simplification 49.0%

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

Alternative 9: 34.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Taylor expanded in x.im around inf 48.9%

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

5. Applied egg-rr 48.9%

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

7. Applied egg-rr 33.2%

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

   1. distribute-rgt-neg-in 33.2%

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

   2. metadata-eval 33.2%

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

   3. *-commutative 33.2%

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

   4. distribute-rgt1-in 33.2%

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

   5. metadata-eval 33.2%

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

   6. neg-mul-1 33.2%

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

9. Simplified 33.2%

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

10. Final simplification 33.2%

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

Alternative 10: 19.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. difference-of-squares 87.9%

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

   2. +-commutative 87.9%

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

4. Applied egg-rr 87.9%

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

6. Simplified 52.0%

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

7. Taylor expanded in x.re around 0 27.2%

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

   1. *-commutative 27.2%

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

9. Simplified 27.2%

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

10. Taylor expanded in x.im around 0 20.8%

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

    1. *-commutative 20.8%

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

    2. *-commutative 20.8%

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

    3. *-commutative 20.8%

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

    4. associate-*r* 20.8%

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

12. Simplified 20.8%

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

Alternative 11: 19.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. difference-of-squares 87.9%

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

   2. +-commutative 87.9%

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

4. Applied egg-rr 87.9%

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

6. Simplified 52.0%

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

7. Taylor expanded in x.re around 0 27.2%

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

   1. *-commutative 27.2%

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

9. Simplified 27.2%

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

10. Taylor expanded in x.im around 0 20.8%

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

11. Final simplification 20.8%

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

Developer Target 1: 87.1% 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 2024141 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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