math.cube on complex, real part

Percentage Accurate: 82.1% → 99.8%

Time: 9.6s

Alternatives: 12

Speedup: 0.9×

• 45.2% 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: 82.1% 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, 0.1× speedup

Math

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

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 2e+99)
    (fma (* x.re_m x.im) (* x.im -3.0) (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 <= 2e+99) {
		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_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 <= 2e+99)
		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(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

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, 2e+99], 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[(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 < 1.9999999999999999e99

   1. Initial program 86.2%

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

   3. Simplified 84.9%

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

      1. associate-*r* 84.9%

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

      2. associate-*l* 84.9%

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

      3. +-commutative 84.9%

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

      4. associate-*r* 90.8%

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

      5. associate-*r* 90.8%

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

      6. fma-define 92.2%

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

   6. Applied egg-rr 92.2%

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

   if 1.9999999999999999e99 < x.re

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. *-un-lft-identity 71.4%

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

      4. distribute-rgt-out 71.4%

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

      5. metadata-eval 71.4%

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

   4. Applied egg-rr 71.4%

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

      1. *-commutative 71.4%

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

      2. count-2 71.4%

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

      3. *-commutative 71.4%

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

      4. add-log-exp 54.3%

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

      5. *-commutative 54.3%

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

      6. exp-prod 60.0%

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

      7. *-commutative 60.0%

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

      8. add-sqr-sqrt 60.0%

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

      9. sqrt-prod 60.0%

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

      10. sqr-neg 60.0%

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

      11. sqrt-unprod 48.6%

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

      12. add-sqr-sqrt 71.4%

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

      13. cancel-sign-sub-inv 71.4%

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

      14. +-inverses 85.7%

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

      15. metadata-eval 85.7%

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

      16. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. --rgt-identity 85.7%

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

      2. difference-of-squares 100.0%

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

      3. associate-*l* 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 93.2%

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

Alternative 2: 93.9% accurate, 0.2× speedup

Math

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

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 2e+99)
    (+ (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 <= 2e+99) {
		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 <= 2d+99) 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 <= 2e+99) {
		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 <= 2e+99:
		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 <= 2e+99)
		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 <= 2e+99)
		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, 2e+99], 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 < 1.9999999999999999e99

   1. Initial program 86.2%

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

   3. Simplified 84.9%

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

   if 1.9999999999999999e99 < x.re

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. *-un-lft-identity 71.4%

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

      4. distribute-rgt-out 71.4%

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

      5. metadata-eval 71.4%

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

   4. Applied egg-rr 71.4%

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

      1. *-commutative 71.4%

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

      2. count-2 71.4%

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

      3. *-commutative 71.4%

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

      4. add-log-exp 54.3%

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

      5. *-commutative 54.3%

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

      6. exp-prod 60.0%

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

      7. *-commutative 60.0%

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

      8. add-sqr-sqrt 60.0%

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

      9. sqrt-prod 60.0%

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

      10. sqr-neg 60.0%

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

      11. sqrt-unprod 48.6%

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

      12. add-sqr-sqrt 71.4%

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

      13. cancel-sign-sub-inv 71.4%

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

      14. +-inverses 85.7%

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

      15. metadata-eval 85.7%

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

      16. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. --rgt-identity 85.7%

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

      2. difference-of-squares 100.0%

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

      3. associate-*l* 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 87.0%

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

Alternative 3: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right)\\ \end{array} \end{array} \]

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 4e+98)
    (-
     (* 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.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 <= 4e+98) {
		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_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 <= 4d+98) 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_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 <= 4e+98) {
		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_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 <= 4e+98:
		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_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 <= 4e+98)
		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(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 <= 4e+98)
		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_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, 4e+98], 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[(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 < 3.99999999999999999e98

   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. *-commutative 86.2%

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

      2. *-un-lft-identity 86.2%

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

      3. *-un-lft-identity 86.2%

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

      4. distribute-rgt-out 86.2%

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

      5. metadata-eval 86.2%

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

   4. Applied egg-rr 86.2%

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

   if 3.99999999999999999e98 < x.re

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. *-un-lft-identity 71.4%

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

      4. distribute-rgt-out 71.4%

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

      5. metadata-eval 71.4%

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

   4. Applied egg-rr 71.4%

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

      1. *-commutative 71.4%

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

      2. count-2 71.4%

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

      3. *-commutative 71.4%

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

      4. add-log-exp 54.3%

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

      5. *-commutative 54.3%

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

      6. exp-prod 60.0%

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

      7. *-commutative 60.0%

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

      8. add-sqr-sqrt 60.0%

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

      9. sqrt-prod 60.0%

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

      10. sqr-neg 60.0%

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

      11. sqrt-unprod 48.6%

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

      12. add-sqr-sqrt 71.4%

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

      13. cancel-sign-sub-inv 71.4%

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

      14. +-inverses 85.7%

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

      15. metadata-eval 85.7%

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

      16. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. --rgt-identity 85.7%

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

      2. difference-of-squares 100.0%

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

      3. associate-*l* 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;x.re\_m - x.re\_m\\ \mathbf{elif}\;x.re\_m \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 2.3e-104)
    (- x.re_m x.re_m)
    (if (<= x.re_m 1.05e+154) (* x.im (* x.re_m -27.0)) (* 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 <= 2.3e-104) {
		tmp = x_46_re_m - x_46_re_m;
	} else if (x_46_re_m <= 1.05e+154) {
		tmp = x_46_im * (x_46_re_m * -27.0);
	} 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 <= 2.3d-104) then
        tmp = x_46re_m - x_46re_m
    else if (x_46re_m <= 1.05d+154) then
        tmp = x_46im * (x_46re_m * (-27.0d0))
    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 <= 2.3e-104) {
		tmp = x_46_re_m - x_46_re_m;
	} else if (x_46_re_m <= 1.05e+154) {
		tmp = x_46_im * (x_46_re_m * -27.0);
	} 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 <= 2.3e-104:
		tmp = x_46_re_m - x_46_re_m
	elif x_46_re_m <= 1.05e+154:
		tmp = x_46_im * (x_46_re_m * -27.0)
	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 <= 2.3e-104)
		tmp = Float64(x_46_re_m - x_46_re_m);
	elseif (x_46_re_m <= 1.05e+154)
		tmp = Float64(x_46_im * Float64(x_46_re_m * -27.0));
	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 <= 2.3e-104)
		tmp = x_46_re_m - x_46_re_m;
	elseif (x_46_re_m <= 1.05e+154)
		tmp = x_46_im * (x_46_re_m * -27.0);
	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, 2.3e-104], N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.05e+154], N[(x$46$im * N[(x$46$re$95$m * -27.0), $MachinePrecision]), $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 < 2.2999999999999999e-104

   1. Initial program 83.0%

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

   3. Simplified 81.3%

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

      1. associate-*r* 81.3%

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

      2. associate-*l* 81.2%

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

      3. +-commutative 81.2%

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

      4. associate-*r* 88.2%

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

      5. associate-*r* 88.3%

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

      6. fma-define 90.0%

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

   6. Applied egg-rr 90.0%

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

   8. Applied egg-rr 23.1%

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

   if 2.2999999999999999e-104 < x.re < 1.04999999999999997e154

   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. difference-of-squares 96.7%

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

   4. Applied egg-rr 96.7%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in x.re around 0 32.5%

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

      1. *-commutative 32.5%

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

   9. Simplified 32.5%

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

       1. associate-*l* 32.5%

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

       2. *-commutative 32.5%

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

       3. distribute-lft-out-- 35.7%

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

       4. *-commutative 35.7%

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

       5. *-commutative 35.7%

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

       6. add-sqr-sqrt 35.7%

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

       7. sqrt-prod 35.7%

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

       8. sqr-neg 35.7%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 3.4%

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

       11. cancel-sign-sub-inv 3.4%

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

       12. +-inverses 6.6%

           \[\leadsto x.im \cdot \left(x.re \cdot -27 - \color{blue}{0}\right) \]

   11. Applied egg-rr 6.6%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot -27 - 0\right)} \]

   12. Taylor expanded in x.im around 0 6.6%

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

       1. *-commutative 6.6%

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

       2. *-commutative 6.6%

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

       3. *-commutative 6.6%

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

       4. associate-*l* 6.6%

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

   14. Simplified 6.6%

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

   if 1.04999999999999997e154 < x.re

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*r* 57.1%

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

      2. associate-*l* 57.1%

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

      3. +-commutative 57.1%

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

      4. associate-*r* 57.1%

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

      5. associate-*r* 57.1%

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

      6. fma-define 61.9%

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

   6. Applied egg-rr 61.9%

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

   8. Applied egg-rr 95.2%

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

Alternative 5: 39.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.6 \cdot 10^{-104}:\\ \;\;\;\;x.re\_m - x.re\_m\\ \mathbf{elif}\;x.re\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 2.6e-104)
    (- x.re_m x.re_m)
    (if (<= x.re_m 1.32e+154) (* (* x.re_m x.im) -27.0) (* 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 <= 2.6e-104) {
		tmp = x_46_re_m - x_46_re_m;
	} else if (x_46_re_m <= 1.32e+154) {
		tmp = (x_46_re_m * x_46_im) * -27.0;
	} 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 <= 2.6d-104) then
        tmp = x_46re_m - x_46re_m
    else if (x_46re_m <= 1.32d+154) then
        tmp = (x_46re_m * x_46im) * (-27.0d0)
    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 <= 2.6e-104) {
		tmp = x_46_re_m - x_46_re_m;
	} else if (x_46_re_m <= 1.32e+154) {
		tmp = (x_46_re_m * x_46_im) * -27.0;
	} 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 <= 2.6e-104:
		tmp = x_46_re_m - x_46_re_m
	elif x_46_re_m <= 1.32e+154:
		tmp = (x_46_re_m * x_46_im) * -27.0
	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 <= 2.6e-104)
		tmp = Float64(x_46_re_m - x_46_re_m);
	elseif (x_46_re_m <= 1.32e+154)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * -27.0);
	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 <= 2.6e-104)
		tmp = x_46_re_m - x_46_re_m;
	elseif (x_46_re_m <= 1.32e+154)
		tmp = (x_46_re_m * x_46_im) * -27.0;
	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, 2.6e-104], N[(x$46$re$95$m - x$46$re$95$m), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.32e+154], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * -27.0), $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 < 2.60000000000000003e-104

   1. Initial program 83.0%

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

   3. Simplified 81.3%

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

      1. associate-*r* 81.3%

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

      2. associate-*l* 81.2%

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

      3. +-commutative 81.2%

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

      4. associate-*r* 88.2%

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

      5. associate-*r* 88.3%

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

      6. fma-define 90.0%

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

   6. Applied egg-rr 90.0%

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

   8. Applied egg-rr 23.1%

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

   if 2.60000000000000003e-104 < x.re < 1.31999999999999998e154

   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. difference-of-squares 96.7%

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

   4. Applied egg-rr 96.7%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in x.re around 0 32.5%

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

      1. *-commutative 32.5%

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

   9. Simplified 32.5%

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

       1. associate-*l* 32.5%

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

       2. *-commutative 32.5%

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

       3. distribute-lft-out-- 35.7%

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

       4. *-commutative 35.7%

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

       5. *-commutative 35.7%

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

       6. add-sqr-sqrt 35.7%

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

       7. sqrt-prod 35.7%

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

       8. sqr-neg 35.7%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 3.4%

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

       11. cancel-sign-sub-inv 3.4%

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

       12. +-inverses 6.6%

           \[\leadsto x.im \cdot \left(x.re \cdot -27 - \color{blue}{0}\right) \]

   11. Applied egg-rr 6.6%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot -27 - 0\right)} \]

   12. Taylor expanded in x.im around 0 6.6%

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

   if 1.31999999999999998e154 < x.re

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*r* 57.1%

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

      2. associate-*l* 57.1%

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

      3. +-commutative 57.1%

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

      4. associate-*r* 57.1%

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

      5. associate-*r* 57.1%

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

      6. fma-define 61.9%

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

   6. Applied egg-rr 61.9%

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

   8. Applied egg-rr 95.2%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{-104}:\\ \;\;\;\;x.re - x.re\\ \mathbf{elif}\;x.re \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -2 + -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 1.32e+154)
    (* x.im (* x.re_m (+ (* x.im -2.0) -27.0)))
    (* 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 <= 1.32e+154) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -2.0) + -27.0));
	} 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 <= 1.32d+154) then
        tmp = x_46im * (x_46re_m * ((x_46im * (-2.0d0)) + (-27.0d0)))
    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 <= 1.32e+154) {
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -2.0) + -27.0));
	} 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 <= 1.32e+154:
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -2.0) + -27.0))
	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 <= 1.32e+154)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(Float64(x_46_im * -2.0) + -27.0)));
	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 <= 1.32e+154)
		tmp = x_46_im * (x_46_re_m * ((x_46_im * -2.0) + -27.0));
	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, 1.32e+154], N[(x$46$im * N[(x$46$re$95$m * N[(N[(x$46$im * -2.0), $MachinePrecision] + -27.0), $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 < 1.31999999999999998e154

   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 88.7%

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

   4. Applied egg-rr 88.7%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in x.re around 0 33.0%

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

      1. *-commutative 33.0%

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

   9. Simplified 33.0%

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

   10. Taylor expanded in x.im around 0 35.9%

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

       1. +-commutative 35.9%

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

       2. associate-*r* 35.9%

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

       3. metadata-eval 35.9%

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

       4. distribute-lft-neg-in 35.9%

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

       5. *-commutative 35.9%

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

       6. distribute-rgt-out 35.9%

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

       7. distribute-rgt-neg-in 35.9%

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

       8. metadata-eval 35.9%

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

   12. Simplified 35.9%

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

   if 1.31999999999999998e154 < x.re

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*r* 57.1%

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

      2. associate-*l* 57.1%

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

      3. +-commutative 57.1%

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

      4. associate-*r* 57.1%

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

      5. associate-*r* 57.1%

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

      6. fma-define 61.9%

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

   6. Applied egg-rr 61.9%

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

   8. Applied egg-rr 95.2%

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

Alternative 7: 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 84.2%

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

   1. *-commutative 84.2%

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

   2. *-un-lft-identity 84.2%

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

   3. *-un-lft-identity 84.2%

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

   4. distribute-rgt-out 84.2%

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

   5. metadata-eval 84.2%

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

4. Applied egg-rr 84.2%

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

   1. *-commutative 84.2%

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

   2. count-2 84.2%

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

   3. *-commutative 84.2%

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

   4. add-log-exp 59.5%

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

   5. *-commutative 59.5%

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

   6. exp-prod 61.5%

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

   7. *-commutative 61.5%

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

   8. add-sqr-sqrt 55.7%

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

   9. sqrt-prod 63.6%

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

   10. sqr-neg 63.6%

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

   11. sqrt-unprod 51.6%

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

   12. add-sqr-sqrt 66.7%

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

   13. cancel-sign-sub-inv 66.7%

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

   14. +-inverses 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

6. Applied egg-rr 71.8%

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

   1. --rgt-identity 71.8%

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

   2. difference-of-squares 78.4%

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

   3. associate-*l* 79.0%

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

8. Applied egg-rr 79.0%

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

9. Final simplification 79.0%

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

Alternative 8: 38.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;x.re\_m - x.re\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

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 7.6e-65) (- 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 <= 7.6e-65) {
		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 <= 7.6d-65) 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 <= 7.6e-65) {
		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 <= 7.6e-65:
		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 <= 7.6e-65)
		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 <= 7.6e-65)
		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, 7.6e-65], 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 < 7.6000000000000003e-65

   1. Initial program 83.6%

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

   3. Simplified 82.0%

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

      1. associate-*r* 82.0%

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

      2. associate-*l* 81.9%

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

      3. +-commutative 81.9%

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

      4. associate-*r* 89.0%

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

      5. associate-*r* 89.0%

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

      6. fma-define 90.6%

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

   6. Applied egg-rr 90.6%

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

   8. Applied egg-rr 21.8%

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

   if 7.6000000000000003e-65 < x.re

   1. Initial program 85.8%

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

   3. Simplified 81.6%

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

      1. associate-*r* 81.6%

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

      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-*r* 81.6%

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

      5. associate-*r* 81.6%

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

      6. fma-define 84.5%

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

   6. Applied egg-rr 84.5%

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

   8. Applied egg-rr 31.3%

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

Alternative 9: 35.3% 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 84.2%

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

3. Simplified 81.9%

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

   1. associate-*r* 81.9%

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

   2. associate-*l* 81.9%

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

   3. +-commutative 81.9%

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

   4. associate-*r* 87.0%

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

   5. associate-*r* 87.0%

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

   6. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 22.6%

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

Alternative 10: 3.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.re\_m \cdot -0.3333333333333333\right) \end{array} \]

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 -0.3333333333333333)))

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 * -0.3333333333333333);
}

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 * (-0.3333333333333333d0))
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 * -0.3333333333333333);
}

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 * -0.3333333333333333)

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 * -0.3333333333333333))
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 * -0.3333333333333333);
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 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 81.9%

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

   1. associate-*r* 81.9%

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

   2. associate-*l* 81.9%

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

   3. +-commutative 81.9%

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

   4. associate-*r* 87.0%

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

   5. associate-*r* 87.0%

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

   6. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 3.1%

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

   1. *-rgt-identity 3.1%

      \[\leadsto \frac{\color{blue}{x.re \cdot 1}}{-3} \]

   2. associate-/l* 3.1%

      \[\leadsto \color{blue}{x.re \cdot \frac{1}{-3}} \]

   3. metadata-eval 3.1%

      \[\leadsto x.re \cdot \color{blue}{-0.3333333333333333} \]

10. Simplified 3.1%

    \[\leadsto \color{blue}{x.re \cdot -0.3333333333333333} \]
11. Add Preprocessing

Alternative 11: 3.0% 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 84.2%

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

3. Simplified 81.9%

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

   1. associate-*r* 81.9%

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

   2. associate-*l* 81.9%

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

   3. +-commutative 81.9%

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

   4. associate-*r* 87.0%

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

   5. associate-*r* 87.0%

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

   6. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.5%

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

   1. sub-neg 2.5%

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

10. Simplified 2.5%

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

11. Taylor expanded in x.re around inf 3.1%

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

    1. mul-1-neg 3.1%

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

13. Simplified 3.1%

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

Alternative 12: 2.7% accurate, 19.0× speedup

Math

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

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 -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) {
	return x_46_re_s * -3.0;
}

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 * (-3.0d0)
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 * -3.0;
}

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 * -3.0

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 * -3.0)
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 * -3.0;
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 * -3.0), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 81.9%

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

   1. associate-*r* 81.9%

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

   2. associate-*l* 81.9%

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

   3. +-commutative 81.9%

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

   4. associate-*r* 87.0%

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

   5. associate-*r* 87.0%

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

   6. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.5%

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

   1. sub-neg 2.5%

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

10. Simplified 2.5%

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

11. Taylor expanded in x.re around 0 2.8%

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

Developer Target 1: 86.4% 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 2024147 
(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)))