math.cube on complex, real part

Percentage Accurate: 82.8% → 96.8%

Time: 7.2s

Alternatives: 5

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot x.im\_m\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1.95e+152)
   (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
   (* (* (* x.re x.im_m) x.im_m) -3.0)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.95e+152) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.95e+152)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * x_46_im_m) * -3.0);
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 1.95e+152], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.95000000000000006e152

   1. Initial program 88.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 1.95000000000000006e152 < x.im

   1. Initial program 37.2%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 94.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot x.im\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -1e-318)
   (* (* (* x.re x.im_m) x.im_m) -3.0)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318) {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-1d-318)) then
        tmp = ((x_46re * x_46im_m) * x_46im_m) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318) {
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318:
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * x_46_im_m) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318)
		tmp = ((x_46_re * x_46_im_m) * x_46_im_m) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-318], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999875e-319

   1. Initial program 93.2%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 54.7%

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

   if -9.9999875e-319 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 76.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x.re around inf

      \[\leadsto {x.re}^{2} \cdot x.re \]
   7. Step-by-step derivation

   8. Applied rewrites 64.7%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.im\_m\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -1e-318)
   (* (* (* x.im_m x.im_m) x.re) -3.0)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318) {
		tmp = ((x_46_im_m * x_46_im_m) * x_46_re) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-1d-318)) then
        tmp = ((x_46im_m * x_46im_m) * x_46re) * (-3.0d0)
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318) {
		tmp = ((x_46_im_m * x_46_im_m) * x_46_re) * -3.0;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318:
		tmp = ((x_46_im_m * x_46_im_m) * x_46_re) * -3.0
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_im_m) * x_46_re) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -1e-318)
		tmp = ((x_46_im_m * x_46_im_m) * x_46_re) * -3.0;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-318], N[(N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999875e-319

   1. Initial program 93.2%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-out-- N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -9.9999875e-319 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 76.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x.re around inf

      \[\leadsto {x.re}^{2} \cdot x.re \]
   7. Step-by-step derivation

   8. Applied rewrites 64.7%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-239}:\\ \;\;\;\;\left(\left(2 - x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-239)
   (* (* (- 2.0 x.im_m) x.re) x.im_m)
   (* (* x.re x.re) x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-239) {
		tmp = ((2.0 - x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-239)) then
        tmp = ((2.0d0 - x_46im_m) * x_46re) * x_46im_m
    else
        tmp = (x_46re * x_46re) * x_46re
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-239) {
		tmp = ((2.0 - x_46_im_m) * x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-239:
		tmp = ((2.0 - x_46_im_m) * x_46_re) * x_46_im_m
	else:
		tmp = (x_46_re * x_46_re) * x_46_re
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-239)
		tmp = Float64(Float64(Float64(2.0 - x_46_im_m) * x_46_re) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-239)
		tmp = ((2.0 - x_46_im_m) * x_46_re) * x_46_im_m;
	else
		tmp = (x_46_re * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-239], N[(N[(N[(2.0 - x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.0000000000000003e-239

   1. Initial program 92.8%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 69.5

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

   5. Applied rewrites 69.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 69.5

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

   7. Applied rewrites 69.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in x.re around 0

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

       1. distribute-rgt-in N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-lft-in N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. associate-*r* N/A

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

       14. distribute-rgt-out N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. mul-1-neg N/A

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

       18. unsub-neg N/A

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

       19. lower--.f64 25.5

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

   12. Applied rewrites 25.5%

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

   if -4.0000000000000003e-239 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 77.2%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out-- N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x.re around inf

      \[\leadsto {x.re}^{2} \cdot x.re \]
   7. Step-by-step derivation

   8. Applied rewrites 63.6%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-239}:\\ \;\;\;\;\left(\left(2 - x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \left(x.re \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 (* (* x.re x.re) x.re))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = (x_46re * x_46re) * x_46re
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return (x_46_re * x_46_re) * x_46_re;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return (x_46_re * x_46_re) * x_46_re

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(Float64(x_46_re * x_46_re) * x_46_re)
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = (x_46_re * x_46_re) * x_46_re;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. distribute-rgt-out-- N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 87.0

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

5. Applied rewrites 87.0%

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

6. Taylor expanded in x.re around inf

   \[\leadsto {x.re}^{2} \cdot x.re \]
7. Step-by-step derivation

8. Applied rewrites 59.0%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
9. Add Preprocessing

Developer Target 1: 87.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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