math.cube on complex, real part

Percentage Accurate: 82.5% → 99.8%

Time: 6.9s

Alternatives: 5

Speedup: 0.9×

• 45.0% 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: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re\_m \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m - x.re\_m\right)\right) + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

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 3.1e+91)
    (+
     (* x.im (+ (* -3.0 (* x.re_m x.im)) (* x.re_m (- x.re_m x.re_m))))
     (pow x.re_m 3.0))
    (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im))))))

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 <= 3.1e+91) {
		tmp = (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m)))) + pow(x_46_re_m, 3.0);
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 3.1d+91) then
        tmp = (x_46im * (((-3.0d0) * (x_46re_m * x_46im)) + (x_46re_m * (x_46re_m - x_46re_m)))) + (x_46re_m ** 3.0d0)
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))
    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 <= 3.1e+91) {
		tmp = (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m)))) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 3.1e+91:
		tmp = (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m)))) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))
	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 <= 3.1e+91)
		tmp = Float64(Float64(x_46_im * Float64(Float64(-3.0 * Float64(x_46_re_m * x_46_im)) + Float64(x_46_re_m * Float64(x_46_re_m - x_46_re_m)))) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)));
	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 <= 3.1e+91)
		tmp = (x_46_im * ((-3.0 * (x_46_re_m * x_46_im)) + (x_46_re_m * (x_46_re_m - x_46_re_m)))) + (x_46_re_m ^ 3.0);
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	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, 3.1e+91], N[(N[(x$46$im * N[(N[(-3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.09999999999999998e91

   1. Initial program 86.6%

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

      1. difference-of-squares 87.0%

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

      2. *-commutative 87.0%

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

   4. Applied egg-rr 87.0%

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

   5. Taylor expanded in x.im around 0 92.2%

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

   6. Taylor expanded in x.re around 0 92.2%

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

   if 3.09999999999999998e91 < x.re

   1. Initial program 84.4%

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

      1. difference-of-squares 86.7%

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

      2. *-commutative 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. associate-*l* 86.7%

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

      2. fma-neg 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-lft-neg-in 86.7%

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

      5. *-commutative 86.7%

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

      6. flip-+ 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. metadata-eval 0.0%

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

      11. distribute-neg-frac 0.0%

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

      12. metadata-eval 0.0%

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

      13. metadata-eval 0.0%

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

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 77.8%

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

      18. *-commutative 77.8%

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

      19. add-log-exp 75.7%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-rgt-identity 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right) + x.re \cdot \left(x.re - x.re\right)\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+91)
    (+ (pow x.re_m 3.0) (* -3.0 (* x.im (* x.re_m x.im))))
    (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+91) {
		tmp = pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1d+91) then
        tmp = (x_46re_m ** 3.0d0) + ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))
    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 <= 1e+91) {
		tmp = Math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1e+91:
		tmp = math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+91)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)));
	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 <= 1e+91)
		tmp = (x_46_re_m ^ 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	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, 1e+91], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000008e91

   1. Initial program 86.6%

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

   3. Simplified 85.7%

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

      1. add-sqr-sqrt 52.9%

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

      2. pow2 52.9%

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

      3. *-commutative 52.9%

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

      4. sqrt-prod 33.4%

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

      5. sqrt-prod 18.8%

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

      6. add-sqr-sqrt 37.3%

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

   6. Applied egg-rr 37.3%

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

      1. unpow2 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-*r* 37.3%

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

      4. associate-*r* 37.3%

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

      5. add-sqr-sqrt 92.2%

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

      6. *-commutative 92.2%

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

   8. Applied egg-rr 92.2%

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

   if 1.00000000000000008e91 < x.re

   1. Initial program 84.4%

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

      1. difference-of-squares 86.7%

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

      2. *-commutative 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. associate-*l* 86.7%

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

      2. fma-neg 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-lft-neg-in 86.7%

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

      5. *-commutative 86.7%

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

      6. flip-+ 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. metadata-eval 0.0%

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

      11. distribute-neg-frac 0.0%

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

      12. metadata-eval 0.0%

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

      13. metadata-eval 0.0%

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

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 77.8%

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

      18. *-commutative 77.8%

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

      19. add-log-exp 75.7%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-rgt-identity 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 93.6%

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

Alternative 3: 94.0% accurate, 0.2× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+91)
    (+ (pow x.re_m 3.0) (* x.re_m (* x.im (* x.im -3.0))))
    (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+91) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1d+91) then
        tmp = (x_46re_m ** 3.0d0) + (x_46re_m * (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))
    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 <= 1e+91) {
		tmp = Math.pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1e+91:
		tmp = math.pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+91)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_re_m * Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)));
	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 <= 1e+91)
		tmp = (x_46_re_m ^ 3.0) + (x_46_re_m * (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	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, 1e+91], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000008e91

   1. Initial program 86.6%

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

   3. Simplified 85.7%

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

   if 1.00000000000000008e91 < x.re

   1. Initial program 84.4%

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

      1. difference-of-squares 86.7%

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

      2. *-commutative 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. associate-*l* 86.7%

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

      2. fma-neg 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-lft-neg-in 86.7%

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

      5. *-commutative 86.7%

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

      6. flip-+ 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. metadata-eval 0.0%

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

      11. distribute-neg-frac 0.0%

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

      12. metadata-eval 0.0%

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

      13. metadata-eval 0.0%

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

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 77.8%

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

      18. *-commutative 77.8%

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

      19. add-log-exp 75.7%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-rgt-identity 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

Alternative 4: 94.0% accurate, 0.9× speedup

Math

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

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+91)
    (-
     (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))
     (* x.im (* (* x.re_m x.im) 2.0)))
    (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+91) {
		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) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1d+91) then
        tmp = (x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))) - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))
    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 <= 1e+91) {
		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) * 2.0));
	} else {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	}
	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 <= 1e+91:
		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) * 2.0))
	else:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+91)
		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) * 2.0)));
	else
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)));
	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 <= 1e+91)
		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) * 2.0));
	else
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im));
	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, 1e+91], 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] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000008e91

   1. Initial program 86.6%

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

      1. difference-of-squares 87.0%

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

      2. *-commutative 87.0%

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

   4. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. *-un-lft-identity 87.0%

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

      3. distribute-lft-in 87.0%

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

      4. distribute-rgt-out 87.0%

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

      5. metadata-eval 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 1.00000000000000008e91 < x.re

   1. Initial program 84.4%

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

      1. difference-of-squares 86.7%

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

      2. *-commutative 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. associate-*l* 86.7%

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

      2. fma-neg 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-lft-neg-in 86.7%

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

      5. *-commutative 86.7%

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

      6. flip-+ 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. metadata-eval 0.0%

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

      11. distribute-neg-frac 0.0%

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

      12. metadata-eval 0.0%

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

      13. metadata-eval 0.0%

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

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 77.8%

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

      18. *-commutative 77.8%

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

      19. add-log-exp 75.7%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-rgt-identity 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

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

Alternative 5: 79.2% accurate, 2.1× speedup

Math

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

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.im) (* x.re_m (+ x.re_m x.im)))))

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_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

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_46im) * (x_46re_m * (x_46re_m + x_46im)))
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_im) * (x_46_re_m * (x_46_re_m + x_46_im)));
}

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_im) * (x_46_re_m * (x_46_re_m + x_46_im)))

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. difference-of-squares 87.0%

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

   2. *-commutative 87.0%

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

4. Applied egg-rr 87.0%

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

   1. associate-*l* 92.3%

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

   2. fma-neg 92.4%

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

   3. *-commutative 92.4%

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

   4. distribute-lft-neg-in 92.4%

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

   5. *-commutative 92.4%

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

   6. flip-+ 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. metadata-eval 0.0%

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

   11. distribute-neg-frac 0.0%

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

   12. metadata-eval 0.0%

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

   13. metadata-eval 0.0%

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

   14. +-inverses 0.0%

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

   15. metadata-eval 0.0%

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

   16. +-inverses 0.0%

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

   17. flip-+ 60.3%

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

   18. *-commutative 60.3%

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

   19. add-log-exp 59.6%

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

6. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)} \]
7. Step-by-step derivation

   1. fma-undefine 79.1%

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

   2. *-commutative 79.1%

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

   3. associate-*l* 78.5%

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

   4. +-rgt-identity 78.5%

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

   5. associate-*l* 79.1%

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

   6. *-commutative 79.1%

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

8. Applied egg-rr 79.1%

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

Developer target: 87.6% 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 2024083 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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