math.cube on complex, real part

Percentage Accurate: 82.2% → 93.9%

Time: 9.3s

Alternatives: 15

Speedup: 0.9×

• 43.6% 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: 82.2% 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: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.im, x.im \cdot -3, {x.re\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 7.6e+152)
    (fma (* x.re_m x.im) (* x.im -3.0) (pow x.re_m 3.0))
    (* 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 <= 7.6e+152) {
		tmp = fma((x_46_re_m * x_46_im), (x_46_im * -3.0), pow(x_46_re_m, 3.0));
	} else {
		tmp = 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 <= 7.6e+152)
		tmp = fma(Float64(x_46_re_m * x_46_im), Float64(x_46_im * -3.0), (x_46_re_m ^ 3.0));
	else
		tmp = Float64(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, 7.6e+152], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.6000000000000001e152

   1. Initial program 85.7%

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

   3. Simplified 84.0%

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

      1. associate-*r* 84.0%

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

      2. associate-*l* 84.0%

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

      3. +-commutative 84.0%

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

      4. associate-*l* 84.0%

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

      5. associate-*r* 84.0%

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

      6. associate-*r* 92.5%

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

      7. fma-define 95.1%

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

   6. Applied egg-rr 95.1%

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

   if 7.6000000000000001e152 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

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

Alternative 2: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot x.re\_m - x.im \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, t\_0, x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<=
         (- (* x.re_m t_0) (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
         2e-147)
      (fma x.re_m t_0 (* x.im (* (* x.re_m x.im) -2.0)))
      (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 t_0 = (x_46_re_m * x_46_re_m) - (x_46_im * x_46_im);
	double tmp;
	if (((x_46_re_m * t_0) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-147) {
		tmp = fma(x_46_re_m, t_0, (x_46_im * ((x_46_re_m * x_46_im) * -2.0)));
	} else {
		tmp = 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)
	t_0 = Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * t_0) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 2e-147)
		tmp = fma(x_46_re_m, t_0, Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * -2.0)));
	else
		tmp = 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_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * t$95$0), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-147], N[(x$46$re$95$m * t$95$0 + N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.9999999999999999e-147

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

      2. sqr-neg 94.3%

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

      3. fmm-def 94.3%

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

      4. sqr-neg 94.3%

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

      5. *-commutative 94.3%

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

      6. distribute-rgt-neg-in 94.3%

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

      7. neg-mul-1 94.3%

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

      8. *-commutative 94.3%

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

      9. distribute-lft-in 94.3%

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

      10. distribute-rgt-out 94.3%

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

      11. metadata-eval 94.3%

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

   3. Simplified 94.3%

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

   if 1.9999999999999999e-147 < (-.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 67.9%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x.re around inf 55.9%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 2 \cdot 10^{-147}:\\ \;\;\;\;t\_0 - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \end{array} \]

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.re_m) (* x.im x.im)))))
   (*
    x.re_s
    (if (<= (- t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))) 2e-147)
      (- t_0 (* x.im (* (* x.re_m x.im) 2.0)))
      (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 t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-147) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    if ((t_0 - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 2d-147) then
        tmp = t_0 - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = x_46re_m ** 3.0d0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-147) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = Math.pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	tmp = 0
	if (t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-147:
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = math.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)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 2e-147)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	tmp = 0.0;
	if ((t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-147)
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-147], N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.9999999999999999e-147

   1. Initial program 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. *-commutative 94.3%

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

      3. *-un-lft-identity 94.3%

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

      4. distribute-rgt-out 94.3%

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

      5. metadata-eval 94.3%

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

   4. Applied egg-rr 94.3%

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

   if 1.9999999999999999e-147 < (-.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 67.9%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x.re around inf 55.9%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-147}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 8.2 \cdot 10^{+155}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 8.2e+155)
    (-
     (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))
     (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
    (* 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 <= 8.2e+155) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	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 <= 8.2d+155) then
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else
        tmp = 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 <= 8.2e+155) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	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 <= 8.2e+155:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_re_m * x_46_re_m
	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 <= 8.2e+155)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(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 <= 8.2e+155)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 8.2e+155], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.1999999999999996e155

   1. Initial program 85.3%

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

      1. difference-of-squares 86.2%

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

      2. *-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

   if 8.1999999999999996e155 < x.re

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. associate-*r* 64.0%

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

      2. associate-*l* 64.0%

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

      3. +-commutative 64.0%

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

      4. associate-*l* 64.0%

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

      5. associate-*r* 64.0%

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

      6. associate-*r* 64.0%

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

      7. fma-define 68.0%

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

   6. Applied egg-rr 68.0%

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

   8. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8.2 \cdot 10^{+155}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 1.35e+154)
    (-
     (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (* x.re_m x.re_m))))

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 <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	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 <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = x_46re_m * x_46re_m
    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 <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	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 <= 1.35e+154:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = x_46_re_m * x_46_re_m
	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 <= 1.35e+154)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = Float64(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 <= 1.35e+154)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.35e+154], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.35000000000000003e154

   1. Initial program 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. *-commutative 85.7%

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

      3. *-un-lft-identity 85.7%

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

      4. distribute-rgt-out 85.7%

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

      5. metadata-eval 85.7%

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

   4. Applied egg-rr 85.7%

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

   if 1.35000000000000003e154 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 550000000:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot -27\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{elif}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 550000000.0)
    (- (* x.re_m (* x.im -27.0)) (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
    (if (<= x.re_m 1.35e+154)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) 0.25)
      (* 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 <= 550000000.0) {
		tmp = (x_46_re_m * (x_46_im * -27.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 550000000.0d0) then
        tmp = (x_46re_m * (x_46im * (-27.0d0))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))
    else if (x_46re_m <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - 0.25d0
    else
        tmp = 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 <= 550000000.0) {
		tmp = (x_46_re_m * (x_46_im * -27.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 550000000.0:
		tmp = (x_46_re_m * (x_46_im * -27.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	elif x_46_re_m <= 1.35e+154:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25
	else:
		tmp = 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 <= 550000000.0)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_im * -27.0)) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - 0.25);
	else
		tmp = Float64(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 <= 550000000.0)
		tmp = (x_46_re_m * (x_46_im * -27.0)) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	else
		tmp = 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[x$46$re$95$m, 550000000.0], N[(N[(x$46$re$95$m * N[(x$46$im * -27.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.35e+154], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 5.5e8

   1. Initial program 84.3%

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

      1. difference-of-squares 84.8%

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

   4. Applied egg-rr 84.8%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in x.re around 0 35.2%

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

      1. associate-*r* 35.2%

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

      2. *-commutative 35.2%

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

   9. Simplified 35.2%

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

   if 5.5e8 < x.re < 1.35000000000000003e154

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.25} \]

   if 1.35000000000000003e154 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 550000000:\\ \;\;\;\;x.re \cdot \left(x.im \cdot -27\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 550000000:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot -27 - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{elif}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 550000000.0)
    (- (* (* x.re_m x.im) -27.0) (* x.im (* (* x.re_m x.im) 2.0)))
    (if (<= x.re_m 1.35e+154)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) 0.25)
      (* 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 <= 550000000.0) {
		tmp = ((x_46_re_m * x_46_im) * -27.0) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 550000000.0d0) then
        tmp = ((x_46re_m * x_46im) * (-27.0d0)) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else if (x_46re_m <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - 0.25d0
    else
        tmp = 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 <= 550000000.0) {
		tmp = ((x_46_re_m * x_46_im) * -27.0) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 550000000.0:
		tmp = ((x_46_re_m * x_46_im) * -27.0) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	elif x_46_re_m <= 1.35e+154:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25
	else:
		tmp = 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 <= 550000000.0)
		tmp = Float64(Float64(Float64(x_46_re_m * x_46_im) * -27.0) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - 0.25);
	else
		tmp = Float64(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 <= 550000000.0)
		tmp = ((x_46_re_m * x_46_im) * -27.0) - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	else
		tmp = 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[x$46$re$95$m, 550000000.0], N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -27.0), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.35e+154], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 5.5e8

   1. Initial program 84.3%

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

      1. difference-of-squares 84.8%

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

   4. Applied egg-rr 84.8%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in x.re around 0 35.2%

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

      1. *-un-lft-identity 84.3%

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

      2. *-commutative 84.3%

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

      3. *-un-lft-identity 84.3%

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

      4. distribute-rgt-out 84.3%

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

      5. metadata-eval 84.3%

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

   9. Applied egg-rr 35.2%

      \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

   if 5.5e8 < x.re < 1.35000000000000003e154

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. *-commutative 96.3%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.25} \]

   if 1.35000000000000003e154 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 550000000:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot -27 - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(-27 - x.im\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - 531441\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 3.4e-9)
    (* x.im (* x.re_m (- -27.0 x.im)))
    (if (<= x.re_m 1.35e+154)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) 531441.0)
      (* 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 <= 3.4e-9) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 531441.0;
	} else {
		tmp = 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 <= 3.4d-9) then
        tmp = x_46im * (x_46re_m * ((-27.0d0) - x_46im))
    else if (x_46re_m <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - 531441.0d0
    else
        tmp = 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 <= 3.4e-9) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 531441.0;
	} else {
		tmp = 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 <= 3.4e-9:
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im))
	elif x_46_re_m <= 1.35e+154:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 531441.0
	else:
		tmp = 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 <= 3.4e-9)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(-27.0 - x_46_im)));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - 531441.0);
	else
		tmp = Float64(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 <= 3.4e-9)
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 531441.0;
	else
		tmp = 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[x$46$re$95$m, 3.4e-9], N[(x$46$im * N[(x$46$re$95$m * N[(-27.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.35e+154], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 531441.0), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 3.3999999999999998e-9

   1. Initial program 84.1%

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

      1. difference-of-squares 84.6%

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

   4. Applied egg-rr 84.6%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in x.re around 0 34.7%

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

      1. *-un-lft-identity 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + x.im \cdot x.re\right) \cdot x.im \]

      2. *-commutative 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(1 \cdot \left(x.re \cdot x.im\right) + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      3. fma-define 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   9. Applied egg-rr 34.7%

      \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   11. Simplified 14.7%

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

   12. Taylor expanded in x.re around inf 33.8%

       \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - {x.im}^{2}\right)} \]
   13. Step-by-step derivation

       1. distribute-rgt-out-- 33.3%

          \[\leadsto \color{blue}{\left(-27 \cdot x.im\right) \cdot x.re - {x.im}^{2} \cdot x.re} \]

       2. *-commutative 33.3%

          \[\leadsto \color{blue}{\left(x.im \cdot -27\right)} \cdot x.re - {x.im}^{2} \cdot x.re \]

       3. associate-*r* 33.8%

          \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re\right)} - {x.im}^{2} \cdot x.re \]

       4. *-commutative 33.8%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot -27\right)} - {x.im}^{2} \cdot x.re \]

       5. unpow2 33.8%

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

       6. associate-*r* 34.5%

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

       7. distribute-lft-out-- 35.0%

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

       8. *-commutative 35.0%

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

       9. distribute-rgt-out-- 35.0%

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

   14. Simplified 35.0%

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

   if 3.3999999999999998e-9 < x.re < 1.35000000000000003e154

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. *-commutative 96.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 97.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{531441} \]

   if 1.35000000000000003e154 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-27 - x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 531441\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(-27 - x.im\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 3.4e-9)
    (* x.im (* x.re_m (- -27.0 x.im)))
    (if (<= x.re_m 1.35e+154)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))) 0.25)
      (* 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 <= 3.4e-9) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 3.4d-9) then
        tmp = x_46im * (x_46re_m * ((-27.0d0) - x_46im))
    else if (x_46re_m <= 1.35d+154) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - 0.25d0
    else
        tmp = 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 <= 3.4e-9) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else if (x_46_re_m <= 1.35e+154) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	} else {
		tmp = 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 <= 3.4e-9:
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im))
	elif x_46_re_m <= 1.35e+154:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25
	else:
		tmp = 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 <= 3.4e-9)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(-27.0 - x_46_im)));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - 0.25);
	else
		tmp = Float64(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 <= 3.4e-9)
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	elseif (x_46_re_m <= 1.35e+154)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - 0.25;
	else
		tmp = 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[x$46$re$95$m, 3.4e-9], N[(x$46$im * N[(x$46$re$95$m * N[(-27.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.35e+154], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 3.3999999999999998e-9

   1. Initial program 84.1%

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

      1. difference-of-squares 84.6%

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

   4. Applied egg-rr 84.6%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in x.re around 0 34.7%

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

      1. *-un-lft-identity 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + x.im \cdot x.re\right) \cdot x.im \]

      2. *-commutative 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(1 \cdot \left(x.re \cdot x.im\right) + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      3. fma-define 34.7%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   9. Applied egg-rr 34.7%

      \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   11. Simplified 14.7%

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

   12. Taylor expanded in x.re around inf 33.8%

       \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - {x.im}^{2}\right)} \]
   13. Step-by-step derivation

       1. distribute-rgt-out-- 33.3%

          \[\leadsto \color{blue}{\left(-27 \cdot x.im\right) \cdot x.re - {x.im}^{2} \cdot x.re} \]

       2. *-commutative 33.3%

          \[\leadsto \color{blue}{\left(x.im \cdot -27\right)} \cdot x.re - {x.im}^{2} \cdot x.re \]

       3. associate-*r* 33.8%

          \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re\right)} - {x.im}^{2} \cdot x.re \]

       4. *-commutative 33.8%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot -27\right)} - {x.im}^{2} \cdot x.re \]

       5. unpow2 33.8%

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

       6. associate-*r* 34.5%

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

       7. distribute-lft-out-- 35.0%

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

       8. *-commutative 35.0%

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

       9. distribute-rgt-out-- 35.0%

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

   14. Simplified 35.0%

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

   if 3.3999999999999998e-9 < x.re < 1.35000000000000003e154

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. *-commutative 96.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 97.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.25} \]

   if 1.35000000000000003e154 < x.re

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

      1. associate-*r* 61.5%

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

      2. associate-*l* 61.5%

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

      3. +-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. associate-*r* 61.5%

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

      6. associate-*r* 61.5%

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

      7. fma-define 65.4%

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

   6. Applied egg-rr 65.4%

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

   8. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-27 - x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.25\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(-27 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 2.7e+154)
    (* x.im (* x.re_m (- -27.0 x.im)))
    (* 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 <= 2.7e+154) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else {
		tmp = 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 <= 2.7d+154) then
        tmp = x_46im * (x_46re_m * ((-27.0d0) - x_46im))
    else
        tmp = 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 <= 2.7e+154) {
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	} else {
		tmp = 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 <= 2.7e+154:
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im))
	else:
		tmp = 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 <= 2.7e+154)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(-27.0 - x_46_im)));
	else
		tmp = Float64(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 <= 2.7e+154)
		tmp = x_46_im * (x_46_re_m * (-27.0 - x_46_im));
	else
		tmp = 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[x$46$re$95$m, 2.7e+154], N[(x$46$im * N[(x$46$re$95$m * N[(-27.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.70000000000000006e154

   1. Initial program 85.3%

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

      1. difference-of-squares 86.2%

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

   4. Applied egg-rr 86.2%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in x.re around 0 35.0%

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

      1. *-un-lft-identity 35.0%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + x.im \cdot x.re\right) \cdot x.im \]

      2. *-commutative 35.0%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \left(1 \cdot \left(x.re \cdot x.im\right) + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      3. fma-define 35.0%

         \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   9. Applied egg-rr 35.0%

      \[\leadsto -27 \cdot \left(x.im \cdot x.re\right) - \color{blue}{\mathsf{fma}\left(1, x.re \cdot x.im, x.re \cdot x.im\right)} \cdot x.im \]

   11. Simplified 17.7%

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

   12. Taylor expanded in x.re around inf 34.7%

       \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - {x.im}^{2}\right)} \]
   13. Step-by-step derivation

       1. distribute-rgt-out-- 33.8%

          \[\leadsto \color{blue}{\left(-27 \cdot x.im\right) \cdot x.re - {x.im}^{2} \cdot x.re} \]

       2. *-commutative 33.8%

          \[\leadsto \color{blue}{\left(x.im \cdot -27\right)} \cdot x.re - {x.im}^{2} \cdot x.re \]

       3. associate-*r* 34.2%

          \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re\right)} - {x.im}^{2} \cdot x.re \]

       4. *-commutative 34.2%

          \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot -27\right)} - {x.im}^{2} \cdot x.re \]

       5. unpow2 34.2%

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

       6. associate-*r* 34.9%

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

       7. distribute-lft-out-- 35.8%

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

       8. *-commutative 35.8%

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

       9. distribute-rgt-out-- 35.8%

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

   14. Simplified 35.8%

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

   if 2.70000000000000006e154 < x.re

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. associate-*r* 64.0%

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

      2. associate-*l* 64.0%

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

      3. +-commutative 64.0%

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

      4. associate-*l* 64.0%

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

      5. associate-*r* 64.0%

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

      6. associate-*r* 64.0%

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

      7. fma-define 68.0%

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

   6. Applied egg-rr 68.0%

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

   8. Applied egg-rr 88.0%

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

Alternative 11: 38.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.2 \cdot 10^{-72}:\\ \;\;\;\;x.re\_m - x.re\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 1.2e-72) (- x.re_m 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 <= 1.2e-72) {
		tmp = x_46_re_m - x_46_re_m;
	} else {
		tmp = 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 <= 1.2d-72) then
        tmp = x_46re_m - x_46re_m
    else
        tmp = 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 <= 1.2e-72) {
		tmp = x_46_re_m - x_46_re_m;
	} else {
		tmp = 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 <= 1.2e-72:
		tmp = x_46_re_m - x_46_re_m
	else:
		tmp = 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 <= 1.2e-72)
		tmp = Float64(x_46_re_m - x_46_re_m);
	else
		tmp = Float64(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 <= 1.2e-72)
		tmp = x_46_re_m - x_46_re_m;
	else
		tmp = 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[x$46$re$95$m, 1.2e-72], N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.2e-72

   1. Initial program 82.7%

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

   3. Simplified 81.1%

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

      1. associate-*r* 81.1%

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

      2. associate-*l* 81.1%

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

      3. +-commutative 81.1%

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

      4. associate-*l* 81.1%

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

      5. associate-*r* 81.1%

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

      6. associate-*r* 91.5%

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

      7. fma-define 94.3%

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

   6. Applied egg-rr 94.3%

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

   8. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{x.re - x.re} \]

   if 1.2e-72 < x.re

   1. Initial program 84.4%

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

   3. Simplified 83.2%

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

      1. associate-*r* 83.2%

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

      2. associate-*l* 83.2%

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

      3. +-commutative 83.2%

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

      4. associate-*l* 83.2%

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

      5. associate-*r* 83.2%

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

      6. associate-*r* 84.4%

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

      7. fma-define 87.0%

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

   6. Applied egg-rr 87.0%

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

   8. Applied egg-rr 32.1%

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

Alternative 12: 35.2% accurate, 6.3× speedup

Math

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

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)))

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);
}

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)
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);
}

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)

Julia

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Simplified 81.7%

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

   1. associate-*r* 81.8%

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

   2. associate-*l* 81.7%

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

   3. +-commutative 81.7%

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

   4. associate-*l* 81.8%

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

   5. associate-*r* 81.7%

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

   6. associate-*r* 89.4%

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

   7. fma-define 92.1%

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

6. Applied egg-rr 92.1%

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

8. Applied egg-rr 25.4%

   \[\leadsto \color{blue}{x.re \cdot x.re} \]
9. Add Preprocessing

Alternative 13: 3.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot -1.3113726523970925 \cdot 10^{-13}\right) \end{array} \]

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 -1.3113726523970925e-13)))

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 * -1.3113726523970925e-13);
}

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 * (-1.3113726523970925d-13))
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 * -1.3113726523970925e-13);
}

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 * -1.3113726523970925e-13)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_re_m * -1.3113726523970925e-13))
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 * -1.3113726523970925e-13);
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(x$46$re$95$m * -1.3113726523970925e-13), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Simplified 81.7%

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

   1. associate-*r* 81.8%

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

   2. associate-*l* 81.7%

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

   3. +-commutative 81.7%

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

   4. associate-*l* 81.8%

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

   5. associate-*r* 81.7%

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

   6. associate-*r* 89.4%

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

   7. fma-define 92.1%

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

6. Applied egg-rr 92.1%

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

8. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{\frac{x.re}{-7625597484987}} \]
9. Step-by-step derivation

   1. *-rgt-identity 3.8%

      \[\leadsto \frac{\color{blue}{x.re \cdot 1}}{-7625597484987} \]

   2. associate-/l* 3.8%

      \[\leadsto \color{blue}{x.re \cdot \frac{1}{-7625597484987}} \]

   3. metadata-eval 3.8%

      \[\leadsto x.re \cdot \color{blue}{-1.3113726523970925 \cdot 10^{-13}} \]

10. Simplified 3.8%

    \[\leadsto \color{blue}{x.re \cdot -1.3113726523970925 \cdot 10^{-13}} \]
11. Add Preprocessing

Alternative 14: 3.1% accurate, 9.5× speedup

Math

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

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)))

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;
}

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
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;
}

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

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(-x_46_re_m))
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;
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 * (-x$46$re$95$m)), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Simplified 81.7%

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

   1. associate-*r* 81.8%

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

   2. associate-*l* 81.7%

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

   3. +-commutative 81.7%

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

   4. associate-*l* 81.8%

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

   5. associate-*r* 81.7%

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

   6. associate-*r* 89.4%

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

   7. fma-define 92.1%

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

6. Applied egg-rr 92.1%

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

8. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{-7625597484987 + \left(-x.re\right)} \]
9. Step-by-step derivation

   1. sub-neg 2.2%

      \[\leadsto \color{blue}{-7625597484987 - x.re} \]

10. Simplified 2.2%

    \[\leadsto \color{blue}{-7625597484987 - x.re} \]

11. Taylor expanded in x.re around inf 3.2%

    \[\leadsto \color{blue}{-1 \cdot x.re} \]
12. Step-by-step derivation

    1. neg-mul-1 3.2%

       \[\leadsto \color{blue}{-x.re} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{-x.re} \]
14. Add Preprocessing

Alternative 15: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot -7625597484987 \end{array} \]

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 -7625597484987.0))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * -7625597484987.0;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (-7625597484987.0d0)
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * -7625597484987.0;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * -7625597484987.0

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * -7625597484987.0)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * -7625597484987.0;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * -7625597484987.0), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Simplified 81.7%

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

   1. associate-*r* 81.8%

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

   2. associate-*l* 81.7%

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

   3. +-commutative 81.7%

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

   4. associate-*l* 81.8%

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

   5. associate-*r* 81.7%

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

   6. associate-*r* 89.4%

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

   7. fma-define 92.1%

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

6. Applied egg-rr 92.1%

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

8. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{-7625597484987 + \left(-x.re\right)} \]
9. Step-by-step derivation

   1. sub-neg 2.2%

      \[\leadsto \color{blue}{-7625597484987 - x.re} \]

10. Simplified 2.2%

    \[\leadsto \color{blue}{-7625597484987 - x.re} \]

11. Taylor expanded in x.re around 0 2.7%

    \[\leadsto \color{blue}{-7625597484987} \]
12. Add Preprocessing

Developer Target 1: 87.4% 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 2024186 
(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)))