math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 99.8%

Time: 8.5s

Alternatives: 9

Speedup: 1.2×

• 43.1% 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: 83.0% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot 3 - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+99)
    (- (* (* x.re (* x.im_m x.re)) 3.0) (pow x.im_m 3.0))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -3.0))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = ((x_46_re * (x_46_im_m * x_46_re)) * 3.0) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+99) then
        tmp = ((x_46re * (x_46im_m * x_46re)) * 3.0d0) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = ((x_46_re * (x_46_im_m * x_46_re)) * 3.0) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+99:
		tmp = ((x_46_re * (x_46_im_m * x_46_re)) * 3.0) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+99)
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_im_m * x_46_re)) * 3.0) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+99)
		tmp = ((x_46_re * (x_46_im_m * x_46_re)) * 3.0) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+99], N[(N[(N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000008e99

   1. Initial program 89.0%

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

   3. Simplified 91.6%

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

   if 5.00000000000000008e99 < x.im

   1. Initial program 54.9%

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

      1. difference-of-squares 66.7%

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

      2. *-commutative 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in x.re around 0 66.7%

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

   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 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+99}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+99)
    (- (* x.re (* x.re (* x.im_m 3.0))) (pow x.im_m 3.0))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -3.0))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+99) then
        tmp = (x_46re * (x_46re * (x_46im_m * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+99:
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+99)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+99)
		tmp = (x_46_re * (x_46_re * (x_46_im_m * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+99], N[(N[(x$46$re * N[(x$46$re * 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[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000008e99

   1. Initial program 89.0%

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

   3. Simplified 91.5%

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

   if 5.00000000000000008e99 < x.im

   1. Initial program 54.9%

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

      1. difference-of-squares 66.7%

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

      2. *-commutative 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in x.re around 0 66.7%

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

   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 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+99}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= ((double) INFINITY)) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -3.0;
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = t_0 + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))
	tmp = 0
	if ((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= math.inf:
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = t_0 + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re)))
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)))) <= Inf)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(t_0 + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re));
	tmp = 0.0;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= Inf)
		tmp = t_0 + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = t_0 + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.8%

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

      1. difference-of-squares 94.8%

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

      2. *-commutative 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. *-commutative 91.2%

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

      2. count-2 91.2%

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

      3. *-commutative 91.2%

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

   6. Applied egg-rr 94.8%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. difference-of-squares 23.5%

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

      2. *-commutative 23.5%

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

   4. Applied egg-rr 23.5%

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

   5. Taylor expanded in x.re around 0 23.5%

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

   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 95.5%

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

Alternative 4: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot \left(x.re - x.im\_m\right) + x.im\_m \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right)\right) + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+99)
    (+
     (+
      (* (* x.im_m x.re) (- x.re x.im_m))
      (* x.im_m (* x.im_m (- x.re x.im_m))))
     (* x.re (* (* x.im_m x.re) 2.0)))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -3.0))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = (((x_46_im_m * x_46_re) * (x_46_re - x_46_im_m)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+99) then
        tmp = (((x_46im_m * x_46re) * (x_46re - x_46im_m)) + (x_46im_m * (x_46im_m * (x_46re - x_46im_m)))) + (x_46re * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+99) {
		tmp = (((x_46_im_m * x_46_re) * (x_46_re - x_46_im_m)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+99:
		tmp = (((x_46_im_m * x_46_re) * (x_46_re - x_46_im_m)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+99)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) * Float64(x_46_re - x_46_im_m)) + Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+99)
		tmp = (((x_46_im_m * x_46_re) * (x_46_re - x_46_im_m)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+99], N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000008e99

   1. Initial program 89.0%

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

      1. difference-of-squares 90.0%

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

      2. *-commutative 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. difference-of-squares 89.0%

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

      3. *-commutative 89.0%

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

      4. difference-of-squares 90.0%

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

      5. *-commutative 90.0%

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

      6. distribute-lft-in 87.1%

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

      7. distribute-rgt-in 84.2%

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

   6. Applied egg-rr 84.2%

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

   7. Taylor expanded in x.im around 0 82.2%

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

   9. Simplified 89.5%

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

       1. *-commutative 89.5%

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

       2. count-2 89.5%

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

       3. *-commutative 89.5%

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

   11. Applied egg-rr 89.5%

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

   if 5.00000000000000008e99 < x.im

   1. Initial program 54.9%

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

      1. difference-of-squares 66.7%

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

      2. *-commutative 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in x.re around 0 66.7%

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

   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 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 9.5 \cdot 10^{-91}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 4.9 \cdot 10^{+60}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right) + x.re \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 9.5e-91)
    (* 3.0 (* x.im_m (* x.re x.re)))
    (if (<= x.im_m 4.9e+60)
      (+
       (* x.im_m (* x.im_m (- x.re x.im_m)))
       (* x.re (* (* x.im_m x.re) 2.0)))
      (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -3.0)))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9.5e-91) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else if (x_46_im_m <= 4.9e+60) {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 9.5d-91) then
        tmp = 3.0d0 * (x_46im_m * (x_46re * x_46re))
    else if (x_46im_m <= 4.9d+60) then
        tmp = (x_46im_m * (x_46im_m * (x_46re - x_46im_m))) + (x_46re * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9.5e-91) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else if (x_46_im_m <= 4.9e+60) {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 9.5e-91:
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re))
	elif x_46_im_m <= 4.9e+60:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 9.5e-91)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re * x_46_re)));
	elseif (x_46_im_m <= 4.9e+60)
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 9.5e-91)
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	elseif (x_46_im_m <= 4.9e+60)
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 9.5e-91], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 4.9e+60], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 9.5e-91

   1. Initial program 86.1%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x.re around inf 60.7%

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

      1. pow2 60.7%

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

   7. Applied egg-rr 60.7%

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

   if 9.5e-91 < x.im < 4.9000000000000003e60

   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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. count-2 99.6%

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

      3. *-commutative 99.6%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x.re around 0 82.8%

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

   if 4.9000000000000003e60 < x.im

   1. Initial program 59.6%

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

      1. difference-of-squares 70.2%

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

      2. *-commutative 70.2%

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

   4. Applied egg-rr 70.2%

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

   5. Taylor expanded in x.re around 0 70.2%

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

   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 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9.5 \cdot 10^{-91}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 4.9 \cdot 10^{+60}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 65:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) + -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 65.0)
    (* 3.0 (* x.im_m (* x.re x.re)))
    (+ (* x.im_m (* (- x.re x.im_m) (+ x.im_m x.re))) -3.0))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 65.0) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 65.0d0) then
        tmp = 3.0d0 * (x_46im_m * (x_46re * x_46re))
    else
        tmp = (x_46im_m * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 65.0) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 65.0:
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re))
	else:
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 65.0)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 65.0)
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	else
		tmp = (x_46_im_m * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 65.0], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 65

   1. Initial program 87.5%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x.re around inf 60.9%

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

      1. pow2 60.9%

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

   7. Applied egg-rr 60.9%

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

   if 65 < x.im

   1. Initial program 70.0%

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

      1. difference-of-squares 77.8%

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

      2. *-commutative 77.8%

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

   4. Applied egg-rr 77.8%

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

   5. Taylor expanded in x.re around 0 77.8%

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

   7. Simplified 98.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 65:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re * x_46_re)));
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re * x_46_re)));
}

Python

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, x_46_im_m):
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re * x_46_re)))

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re * x_46_re))))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * (3.0 * (x_46_im_m * (x_46_re * x_46_re)));
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(3.0 * N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Simplified 83.8%

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

5. Taylor expanded in x.re around inf 48.7%

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

   1. pow2 48.7%

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

7. Applied egg-rr 48.7%

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

Alternative 8: 5.3% accurate, 3.8× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (+ -1.0 (/ x.re x.im_m))))

C

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, double x_46_im_m) {
	return x_46_im_s * (-1.0 + (x_46_re / x_46_im_m));
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * (-1.0 + (x_46_re / x_46_im_m));
}

Python

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, x_46_im_m):
	return x_46_im_s * (-1.0 + (x_46_re / x_46_im_m))

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(-1.0 + Float64(x_46_re / x_46_im_m)))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * (-1.0 + (x_46_re / x_46_im_m));
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(-1.0 + N[(x$46$re / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. difference-of-squares 85.4%

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

   2. *-commutative 85.4%

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

4. Applied egg-rr 85.4%

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

   1. *-commutative 85.4%

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

   2. difference-of-squares 82.2%

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

   3. *-commutative 82.2%

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

   4. difference-of-squares 85.4%

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

   5. *-commutative 85.4%

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

   6. distribute-lft-in 81.9%

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

   7. distribute-rgt-in 76.0%

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

6. Applied egg-rr 76.0%

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

7. Taylor expanded in x.im around 0 73.3%

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

9. Simplified 80.3%

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

10. Taylor expanded in x.im around inf 60.9%

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

12. Simplified 4.7%

    \[\leadsto \color{blue}{-1 + \frac{x.re}{x.im}} \]
13. Add Preprocessing

Alternative 9: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

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, x_46_im_m):
	return x_46_im_s * -3.0

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * -3.0), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x.re around 0 82.2%

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

5. Simplified 52.2%

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

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

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

Developer Target 1: 92.0% 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 2024172 
(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)))