math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 99.6%

Time: 8.0s

Alternatives: 10

Speedup: 1.4×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot 3 - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+82)
    (- (* (* x.re_m (* x.im_m x.re_m)) 3.0) (pow x.im_m 3.0))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+82) {
		tmp = ((x_46_re_m * (x_46_im_m * x_46_re_m)) * 3.0) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+82) then
        tmp = ((x_46re_m * (x_46im_m * x_46re_m)) * 3.0d0) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+82) {
		tmp = ((x_46_re_m * (x_46_im_m * x_46_re_m)) * 3.0) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+82:
		tmp = ((x_46_re_m * (x_46_im_m * x_46_re_m)) * 3.0) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+82)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_im_m * x_46_re_m)) * 3.0) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+82)
		tmp = ((x_46_re_m * (x_46_im_m * x_46_re_m)) * 3.0) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+82], N[(N[(N[(x$46$re$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000015e82

   1. Initial program 85.7%

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

   3. Simplified 88.2%

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

   if 5.00000000000000015e82 < x.im

   1. Initial program 60.0%

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

      1. difference-of-squares 72.0%

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

      2. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x.re around 0 64.0%

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

   6. Taylor expanded in x.re around 0 64.0%

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

   8. Simplified 92.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.1 \cdot 10^{+85}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4.1e+85)
    (- (* x.re_m (* x.re_m (* x.im_m 3.0))) (pow x.im_m 3.0))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4.1e+85) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 4.1d+85) then
        tmp = (x_46re_m * (x_46re_m * (x_46im_m * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4.1e+85) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 4.1e+85:
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4.1e+85)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 4.1e+85)
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.1e+85], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.09999999999999978e85

   1. Initial program 85.7%

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

   3. Simplified 88.2%

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

   if 4.09999999999999978e85 < x.im

   1. Initial program 60.0%

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

      1. difference-of-squares 72.0%

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

      2. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x.re around 0 64.0%

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

   6. Taylor expanded in x.re around 0 64.0%

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

   8. Simplified 92.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.1 \cdot 10^{+85}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;t\_0 + x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 2 \cdot 10^{+84}:\\ \;\;\;\;t\_0 - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (* (* x.im_m x.re_m) 2.0))))
   (*
    x.im_s
    (if (<= x.im_m 4.6e-102)
      (+ t_0 (* x.im_m (* x.re_m (- x.re_m x.im_m))))
      (if (<= x.im_m 2e+84)
        (- t_0 (* x.im_m (* x.im_m x.im_m)))
        (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0);
	double tmp;
	if (x_46_im_m <= 4.6e-102) {
		tmp = t_0 + (x_46_im_m * (x_46_re_m * (x_46_re_m - x_46_im_m)));
	} else if (x_46_im_m <= 2e+84) {
		tmp = t_0 - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0)
    if (x_46im_m <= 4.6d-102) then
        tmp = t_0 + (x_46im_m * (x_46re_m * (x_46re_m - x_46im_m)))
    else if (x_46im_m <= 2d+84) then
        tmp = t_0 - (x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0);
	double tmp;
	if (x_46_im_m <= 4.6e-102) {
		tmp = t_0 + (x_46_im_m * (x_46_re_m * (x_46_re_m - x_46_im_m)));
	} else if (x_46_im_m <= 2e+84) {
		tmp = t_0 - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)
	tmp = 0
	if x_46_im_m <= 4.6e-102:
		tmp = t_0 + (x_46_im_m * (x_46_re_m * (x_46_re_m - x_46_im_m)))
	elif x_46_im_m <= 2e+84:
		tmp = t_0 - (x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0))
	tmp = 0.0
	if (x_46_im_m <= 4.6e-102)
		tmp = Float64(t_0 + Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))));
	elseif (x_46_im_m <= 2e+84)
		tmp = Float64(t_0 - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0);
	tmp = 0.0;
	if (x_46_im_m <= 4.6e-102)
		tmp = t_0 + (x_46_im_m * (x_46_re_m * (x_46_re_m - x_46_im_m)));
	elseif (x_46_im_m <= 2e+84)
		tmp = t_0 - (x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.6e-102], N[(t$95$0 + N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2e+84], N[(t$95$0 - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 4.59999999999999973e-102

   1. Initial program 82.8%

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

      1. difference-of-squares 84.6%

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

      2. *-commutative 84.6%

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in x.im around inf 80.6%

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

      1. *-commutative 82.8%

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

      2. count-2 82.8%

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

      3. *-commutative 82.8%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in x.im around 0 61.8%

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

   if 4.59999999999999973e-102 < x.im < 2.00000000000000012e84

   1. Initial program 99.7%

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

      1. difference-of-squares 99.7%

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

      2. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

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

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

      1. *-commutative 99.7%

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

      2. count-2 99.7%

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

      3. *-commutative 99.7%

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

   7. Applied egg-rr 84.0%

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

   8. Taylor expanded in x.re around 0 85.1%

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

   if 2.00000000000000012e84 < x.im

   1. Initial program 60.0%

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

      1. difference-of-squares 72.0%

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

      2. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x.re around 0 64.0%

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

   6. Taylor expanded in x.re around 0 64.0%

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

   8. Simplified 92.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+84}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;\left(x.im\_m \cdot 3\right) \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{elif}\;x.im\_m \leq 5 \cdot 10^{+82}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right) - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3.2e-103)
    (* (* x.im_m 3.0) (* x.re_m x.re_m))
    (if (<= x.im_m 5e+82)
      (- (* x.re_m (* (* x.im_m x.re_m) 2.0)) (* x.im_m (* x.im_m x.im_m)))
      (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.2e-103) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else if (x_46_im_m <= 5e+82) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 3.2d-103) then
        tmp = (x_46im_m * 3.0d0) * (x_46re_m * x_46re_m)
    else if (x_46im_m <= 5d+82) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0)) - (x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.2e-103) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else if (x_46_im_m <= 5e+82) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 3.2e-103:
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m)
	elif x_46_im_m <= 5e+82:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.2e-103)
		tmp = Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re_m * x_46_re_m));
	elseif (x_46_im_m <= 5e+82)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)) - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 3.2e-103)
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	elseif (x_46_im_m <= 5e+82)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.2e-103], N[(N[(x$46$im$95$m * 3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 5e+82], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.19999999999999976e-103

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. sqr-neg 82.8%

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

      4. fma-define 83.5%

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

      5. remove-double-neg 83.5%

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

      6. *-commutative 83.5%

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

      7. distribute-neg-out 83.5%

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

      8. distribute-lft-neg-out 83.5%

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

      9. distribute-lft-neg-out 83.5%

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

      10. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x.re around inf 58.1%

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

      1. pow2 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. *-un-lft-identity 58.1%

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

      2. distribute-rgt-out 58.1%

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

      3. metadata-eval 58.1%

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

   9. Applied egg-rr 58.1%

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

   if 3.19999999999999976e-103 < x.im < 5.00000000000000015e82

   1. Initial program 99.7%

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

      1. difference-of-squares 99.7%

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

      2. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

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

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

      1. *-commutative 99.7%

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

      2. count-2 99.7%

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

      3. *-commutative 99.7%

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

   7. Applied egg-rr 84.0%

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

   8. Taylor expanded in x.re around 0 85.1%

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

   if 5.00000000000000015e82 < x.im

   1. Initial program 60.0%

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

      1. difference-of-squares 72.0%

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

      2. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x.re around 0 64.0%

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

   6. Taylor expanded in x.re around 0 64.0%

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

   8. Simplified 92.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+82}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{+85}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4e+85)
    (+
     (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
     (* x.re_m (* (* x.im_m x.re_m) 2.0)))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4e+85) {
		tmp = (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 4d+85) then
        tmp = (x_46im_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) + (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4e+85) {
		tmp = (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 4e+85:
		tmp = (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4e+85)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 4e+85)
		tmp = (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4e+85], N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.0000000000000001e85

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. count-2 85.7%

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

      3. *-commutative 85.7%

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

   4. Applied egg-rr 85.7%

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

   if 4.0000000000000001e85 < x.im

   1. Initial program 60.0%

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

      1. difference-of-squares 72.0%

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

      2. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x.re around 0 64.0%

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

   6. Taylor expanded in x.re around 0 64.0%

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

   8. Simplified 92.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{+85}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.8:\\ \;\;\;\;\left(x.im\_m \cdot 3\right) \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 7.8)
    (* (* x.im_m 3.0) (* x.re_m x.re_m))
    (+ -3.0 (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.8) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 7.8d0) then
        tmp = (x_46im_m * 3.0d0) * (x_46re_m * x_46re_m)
    else
        tmp = (-3.0d0) + (x_46im_m * ((x_46re_m - x_46im_m) * (x_46im_m + x_46re_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.8) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 7.8:
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m)
	else:
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m)))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.8)
		tmp = Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(-3.0 + Float64(x_46_im_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_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 7.8)
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	else
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 7.8], N[(N[(x$46$im$95$m * 3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(-3.0 + N[(x$46$im$95$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 x.im < 7.79999999999999982

   1. Initial program 85.0%

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

      1. +-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. sqr-neg 85.0%

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

      4. fma-define 85.5%

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

      5. remove-double-neg 85.5%

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

      6. *-commutative 85.5%

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

      7. distribute-neg-out 85.5%

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

      8. distribute-lft-neg-out 85.5%

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

      9. distribute-lft-neg-out 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in x.re around inf 56.7%

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

      1. pow2 56.7%

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

   7. Applied egg-rr 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. distribute-rgt-out 56.7%

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

      3. metadata-eval 56.7%

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

   9. Applied egg-rr 56.7%

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

   if 7.79999999999999982 < x.im

   1. Initial program 66.6%

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

      1. difference-of-squares 76.6%

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

      2. *-commutative 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in x.re around 0 76.6%

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

   7. Simplified 100.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.8:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 32:\\ \;\;\;\;\left(x.im\_m \cdot 3\right) \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 32.0)
    (* (* x.im_m 3.0) (* x.re_m x.re_m))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) -3.0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 32.0) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 32.0d0) then
        tmp = (x_46im_m * 3.0d0) * (x_46re_m * x_46re_m)
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 32.0) {
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 32.0:
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m)
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 32.0)
		tmp = Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 32.0)
		tmp = (x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m);
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 32.0], N[(N[(x$46$im$95$m * 3.0), $MachinePrecision] * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 32

   1. Initial program 85.0%

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

      1. +-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. sqr-neg 85.0%

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

      4. fma-define 85.5%

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

      5. remove-double-neg 85.5%

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

      6. *-commutative 85.5%

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

      7. distribute-neg-out 85.5%

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

      8. distribute-lft-neg-out 85.5%

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

      9. distribute-lft-neg-out 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in x.re around inf 56.7%

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

      1. pow2 56.7%

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

   7. Applied egg-rr 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. distribute-rgt-out 56.7%

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

      3. metadata-eval 56.7%

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

   9. Applied egg-rr 56.7%

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

   if 32 < x.im

   1. Initial program 66.6%

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

      1. difference-of-squares 76.6%

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

      2. *-commutative 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in x.re around 0 70.0%

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

   6. Taylor expanded in x.re around 0 70.0%

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

   8. Simplified 87.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 32:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46im_m * 3.0d0) * (x_46re_m * x_46re_m))
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m));
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * ((x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m))

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_im_m * 3.0) * Float64(x_46_re_m * x_46_re_m)))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * ((x_46_im_m * 3.0) * (x_46_re_m * x_46_re_m));
end

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. +-commutative 80.7%

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

   2. *-commutative 80.7%

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

   3. sqr-neg 80.7%

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

   4. fma-define 82.7%

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

   5. remove-double-neg 82.7%

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

   6. *-commutative 82.7%

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

   7. distribute-neg-out 82.7%

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

   8. distribute-lft-neg-out 82.7%

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

   9. distribute-lft-neg-out 82.7%

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

   10. *-commutative 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in x.re around inf 48.2%

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

   1. pow2 48.2%

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

7. Applied egg-rr 48.2%

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

   1. *-un-lft-identity 48.2%

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

   2. distribute-rgt-out 48.2%

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

   3. metadata-eval 48.2%

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

9. Applied egg-rr 48.2%

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

10. Final simplification 48.2%

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

Alternative 9: 26.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot 9\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (* x.im_s (* (* x.re_m x.re_m) 9.0)))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * 9.0);
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46re_m * x_46re_m) * 9.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * 9.0);
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * 9.0)

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re_m * x_46_re_m) * 9.0))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * ((x_46_re_m * x_46_re_m) * 9.0);
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

   1. +-commutative 80.7%

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

   2. *-commutative 80.7%

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

   3. sqr-neg 80.7%

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

   4. fma-define 82.7%

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

   5. remove-double-neg 82.7%

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

   6. *-commutative 82.7%

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

   7. distribute-neg-out 82.7%

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

   8. distribute-lft-neg-out 82.7%

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

   9. distribute-lft-neg-out 82.7%

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

   10. *-commutative 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in x.re around inf 48.2%

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

   1. pow2 48.2%

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

7. Applied egg-rr 48.2%

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

   1. flip3-+ 18.9%

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

   2. div-inv 18.9%

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

   3. *-commutative 18.9%

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

   4. unpow-prod-down 18.9%

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

   5. metadata-eval 18.9%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\left({x.im}^{3} + {x.im}^{3} \cdot \color{blue}{8}\right) \cdot \frac{1}{x.im \cdot x.im + \left(\left(2 \cdot x.im\right) \cdot \left(2 \cdot x.im\right) - x.im \cdot \left(2 \cdot x.im\right)\right)}\right) \]

   6. fma-define 18.9%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\left({x.im}^{3} + {x.im}^{3} \cdot 8\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x.im, x.im, \left(2 \cdot x.im\right) \cdot \left(2 \cdot x.im\right) - x.im \cdot \left(2 \cdot x.im\right)\right)}}\right) \]

   7. distribute-rgt-out-- 18.9%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\left({x.im}^{3} + {x.im}^{3} \cdot 8\right) \cdot \frac{1}{\mathsf{fma}\left(x.im, x.im, \color{blue}{\left(2 \cdot x.im\right) \cdot \left(2 \cdot x.im - x.im\right)}\right)}\right) \]

   8. *-commutative 18.9%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\left({x.im}^{3} + {x.im}^{3} \cdot 8\right) \cdot \frac{1}{\mathsf{fma}\left(x.im, x.im, \color{blue}{\left(x.im \cdot 2\right)} \cdot \left(2 \cdot x.im - x.im\right)\right)}\right) \]

   9. *-commutative 18.9%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\left({x.im}^{3} + {x.im}^{3} \cdot 8\right) \cdot \frac{1}{\mathsf{fma}\left(x.im, x.im, \left(x.im \cdot 2\right) \cdot \left(\color{blue}{x.im \cdot 2} - x.im\right)\right)}\right) \]

9. Applied egg-rr 18.9%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(\left({x.im}^{3} + {x.im}^{3} \cdot 8\right) \cdot \frac{1}{\mathsf{fma}\left(x.im, x.im, \left(x.im \cdot 2\right) \cdot \left(x.im \cdot 2 - x.im\right)\right)}\right)} \]

11. Simplified 20.8%

    \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{9} \]
12. Add Preprocessing

Alternative 10: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-3.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -3.0

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. difference-of-squares 84.2%

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

   2. *-commutative 84.2%

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

4. Applied egg-rr 84.2%

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

5. Taylor expanded in x.re around 0 84.1%

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

7. Simplified 58.5%

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

8. Taylor expanded in x.im around 0 2.8%

   \[\leadsto \color{blue}{-3} \]
9. Add Preprocessing

Developer Target 1: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024173 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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