math.cube on complex, real part

Percentage Accurate: 82.6% → 99.7%

Time: 10.7s

Alternatives: 7

Speedup: 0.7×

• 44.4% 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: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+149}:\\ \;\;\;\;\left(x.im \cdot x.re\_m\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.re\_m - x.im\right) \cdot \left(x.im + x.re\_m\right)\right) \cdot x.re\_m - \left(x.re\_m \cdot \left(x.im + x.im\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{x.re\_m}{x.im}}{x.im}, x.re\_m, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 -5e+149)
      (* (* x.im x.re_m) (* x.im -3.0))
      (if (<= t_0 INFINITY)
        (-
         (* (* (- x.re_m x.im) (+ x.im x.re_m)) x.re_m)
         (* (* x.re_m (+ x.im x.im)) x.im))
        (*
         (* (fma (/ (/ x.re_m x.im) x.im) x.re_m -3.0) (* x.im x.im))
         x.re_m))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -5e+149) {
		tmp = (x_46_im * x_46_re_m) * (x_46_im * -3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((x_46_re_m - x_46_im) * (x_46_im + x_46_re_m)) * x_46_re_m) - ((x_46_re_m * (x_46_im + x_46_im)) * x_46_im);
	} else {
		tmp = (fma(((x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * (x_46_im * x_46_im)) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= -5e+149)
		tmp = Float64(Float64(x_46_im * x_46_re_m) * Float64(x_46_im * -3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_im + x_46_re_m)) * x_46_re_m) - Float64(Float64(x_46_re_m * Float64(x_46_im + x_46_im)) * x_46_im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * Float64(x_46_im * x_46_im)) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -5e+149], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$46$re$95$m / x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision] * x$46$re$95$m + -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 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)) < -4.9999999999999999e149

   1. Initial program 83.7%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]
   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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 41.7%

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

   if -4.9999999999999999e149 < (-.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)) < +inf.0

   1. Initial program 91.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 91.7

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

   4. Applied rewrites 91.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 91.7

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

   6. Applied rewrites 91.7%

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

   if +inf.0 < (-.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 0.0%

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

   3. 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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.1

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

   8. Applied rewrites 44.1%

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

   9. Taylor expanded in x.im around inf

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

   11. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{x.re}{x.im}}{x.im}, x.re, -3\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e-10)
    (fma -3.0 (* (* x.im x.re_m) x.im) (pow x.re_m 3.0))
    (* (fma (/ (* -3.0 x.im) x.re_m) (/ x.im x.re_m) 1.0) (pow x.re_m 3.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e-10) {
		tmp = fma(-3.0, ((x_46_im * x_46_re_m) * x_46_im), pow(x_46_re_m, 3.0));
	} else {
		tmp = fma(((-3.0 * x_46_im) / x_46_re_m), (x_46_im / x_46_re_m), 1.0) * pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e-10)
		tmp = fma(-3.0, Float64(Float64(x_46_im * x_46_re_m) * x_46_im), (x_46_re_m ^ 3.0));
	else
		tmp = Float64(fma(Float64(Float64(-3.0 * x_46_im) / x_46_re_m), Float64(x_46_im / x_46_re_m), 1.0) * (x_46_re_m ^ 3.0));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e-10], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-3.0 * x$46$im), $MachinePrecision] / x$46$re$95$m), $MachinePrecision] * N[(x$46$im / x$46$re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000004e-10

   1. Initial program 79.7%

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

   3. Taylor expanded in x.re around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 87.7%

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

   if 1.00000000000000004e-10 < x.re

   1. Initial program 70.1%

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

   3. Taylor expanded in x.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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3 \cdot x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot {x.re}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-3, \left(x.im \cdot x.re\_m\right) \cdot x.im, {x.re\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x.im}{x.re\_m} \cdot -3, \frac{x.im}{x.re\_m}, 1\right) \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+102)
    (fma -3.0 (* (* x.im x.re_m) x.im) (pow x.re_m 3.0))
    (*
     (* (* (fma (* (/ x.im x.re_m) -3.0) (/ x.im x.re_m) 1.0) x.re_m) x.re_m)
     x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+102) {
		tmp = fma(-3.0, ((x_46_im * x_46_re_m) * x_46_im), pow(x_46_re_m, 3.0));
	} else {
		tmp = ((fma(((x_46_im / x_46_re_m) * -3.0), (x_46_im / x_46_re_m), 1.0) * x_46_re_m) * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5e+102)
		tmp = fma(-3.0, Float64(Float64(x_46_im * x_46_re_m) * x_46_im), (x_46_re_m ^ 3.0));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(x_46_im / x_46_re_m) * -3.0), Float64(x_46_im / x_46_re_m), 1.0) * x_46_re_m) * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e+102], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$46$im / x$46$re$95$m), $MachinePrecision] * -3.0), $MachinePrecision] * N[(x$46$im / x$46$re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5e102

   1. Initial program 81.3%

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

   3. Taylor expanded in x.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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 88.6%

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

   if 5e102 < x.re

   1. Initial program 57.5%

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

   3. 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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 75.0

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in x.re around inf

      \[\leadsto \left({x.re}^{2} \cdot \left(1 + -3 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right) \cdot x.re \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x.im}{x.re} \cdot -3, \frac{x.im}{x.re}, 1\right) \cdot x.re\right) \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       0.0)
    (* (* x.im x.re_m) (* x.im -3.0))
    (*
     (* (* (fma (* (/ x.im x.re_m) -3.0) (/ x.im x.re_m) 1.0) x.re_m) x.re_m)
     x.re_m))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 0.0

   1. Initial program 92.0%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 65.5

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

   5. Applied rewrites 65.5%

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

   7. Applied rewrites 65.6%

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

   if 0.0 < (-.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 62.7%

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

   3. Taylor expanded in x.re around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-out N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in x.re around inf

      \[\leadsto \left({x.re}^{2} \cdot \left(1 + -3 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right) \cdot x.re \]
   10. Step-by-step derivation

   11. Applied rewrites 75.8%

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{x.im}{x.re} \cdot -3, \frac{x.im}{x.re}, 1\right) \cdot x.re\right) \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\left(x.im \cdot x.re\_m\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -1e-303)
    (* (* x.im x.re_m) (* x.im -3.0))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303) {
		tmp = (x_46_im * x_46_re_m) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-1d-303)) then
        tmp = (x_46im * x_46re_m) * (x_46im * (-3.0d0))
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303) {
		tmp = (x_46_im * x_46_re_m) * (x_46_im * -3.0);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303:
		tmp = (x_46_im * x_46_re_m) * (x_46_im * -3.0)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303)
		tmp = Float64(Float64(x_46_im * x_46_re_m) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303)
		tmp = (x_46_im * x_46_re_m) * (x_46_im * -3.0);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1e-303], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999931e-304

   1. Initial program 88.5%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 50.8%

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

   if -9.99999999999999931e-304 < (-.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 71.5%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 62.1

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.1%

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

Alternative 6: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.im \leq -1 \cdot 10^{-303}:\\ \;\;\;\;-3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.im))
       -1e-303)
    (* -3.0 (* (* x.im x.re_m) x.im))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46im)) <= (-1d-303)) then
        tmp = (-3.0d0) * ((x_46im * x_46re_m) * x_46im)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303) {
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303:
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303)
		tmp = Float64(-3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_im));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_im)) <= -1e-303)
		tmp = -3.0 * ((x_46_im * x_46_re_m) * x_46_im);
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1e-303], N[(-3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999931e-304

   1. Initial program 88.5%

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

   3. 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. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   if -9.99999999999999931e-304 < (-.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 71.5%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 62.1

         \[\leadsto \color{blue}{{x.re}^{3}} \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.1%

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

Alternative 7: 58.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{{x.re}^{3}} \]
4. Step-by-step derivation

   1. lower-pow.f64 57.8

      \[\leadsto \color{blue}{{x.re}^{3}} \]

5. Applied rewrites 57.8%

   \[\leadsto \color{blue}{{x.re}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 57.7%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{x.re} \]
8. 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 2024321 
(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)))