math.cube on complex, real part

Percentage Accurate: 83.0% → 99.5%

Time: 8.1s

Alternatives: 5

Speedup: 0.9×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+71}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right) + x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 0\right)\right)\right) + {x.re\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+71)
    (+
     (+ (* x.im (* x.im (* x.re_m -3.0))) (* x.im (* x.re_m (* x.re_m 0.0))))
     (pow x.re_m 3.0))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+71) {
		tmp = ((x_46_im * (x_46_im * (x_46_re_m * -3.0))) + (x_46_im * (x_46_re_m * (x_46_re_m * 0.0)))) + pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1d+71) then
        tmp = ((x_46im * (x_46im * (x_46re_m * (-3.0d0)))) + (x_46im * (x_46re_m * (x_46re_m * 0.0d0)))) + (x_46re_m ** 3.0d0)
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+71) {
		tmp = ((x_46_im * (x_46_im * (x_46_re_m * -3.0))) + (x_46_im * (x_46_re_m * (x_46_re_m * 0.0)))) + Math.pow(x_46_re_m, 3.0);
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1e+71:
		tmp = ((x_46_im * (x_46_im * (x_46_re_m * -3.0))) + (x_46_im * (x_46_re_m * (x_46_re_m * 0.0)))) + math.pow(x_46_re_m, 3.0)
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+71)
		tmp = Float64(Float64(Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0))) + Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * 0.0)))) + (x_46_re_m ^ 3.0));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1e+71)
		tmp = ((x_46_im * (x_46_im * (x_46_re_m * -3.0))) + (x_46_im * (x_46_re_m * (x_46_re_m * 0.0)))) + (x_46_re_m ^ 3.0);
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e+71], N[(N[(N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1e71

   1. Initial program 87.9%

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

      1. difference-of-squares 90.3%

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

      2. *-commutative 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in x.im around 0 91.9%

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

      1. distribute-rgt-in 91.9%

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

      2. distribute-rgt-out-- 91.9%

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

      3. metadata-eval 91.9%

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

      4. distribute-rgt1-in 91.9%

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

      5. metadata-eval 91.9%

         \[\leadsto \left(\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im + \left(x.re \cdot \left(\color{blue}{0} \cdot x.re\right)\right) \cdot x.im\right) + {x.re}^{3} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\left(\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im + \left(x.re \cdot \left(0 \cdot x.re\right)\right) \cdot x.im\right)} + {x.re}^{3} \]

   if 1e71 < x.re

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. *-commutative 70.4%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 90.7%

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

      1. difference-of-squares 72.2%

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

      2. *-commutative 72.2%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re - 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+71}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) + x.im \cdot \left(x.re \cdot \left(x.re \cdot 0\right)\right)\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.75 \cdot 10^{+71}:\\ \;\;\;\;{x.re\_m}^{3} + -3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

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 1.75e+71)
    (+ (pow x.re_m 3.0) (* -3.0 (* x.im (* x.re_m x.im))))
    (* x.re_m (* (- x.re_m x.im) (+ 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 <= 1.75e+71) {
		tmp = pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (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 <= 1.75d+71) then
        tmp = (x_46re_m ** 3.0d0) + ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (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 <= 1.75e+71) {
		tmp = Math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (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 <= 1.75e+71:
		tmp = math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (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 <= 1.75e+71)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * 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 <= 1.75e+71)
		tmp = (x_46_re_m ^ 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (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, 1.75e+71], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.75e71

   1. Initial program 87.9%

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

   3. Simplified 85.9%

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

      1. add-sqr-sqrt 53.6%

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

      2. pow2 53.6%

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

      3. *-commutative 53.6%

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

      4. sqrt-prod 39.2%

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

      5. sqrt-prod 21.2%

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

      6. add-sqr-sqrt 42.4%

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

   6. Applied egg-rr 42.4%

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

      1. unpow2 42.4%

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

      2. *-commutative 42.4%

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

      3. associate-*r* 42.4%

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

      4. associate-*r* 42.4%

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

      5. add-sqr-sqrt 91.9%

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

      6. *-commutative 91.9%

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

   8. Applied egg-rr 91.9%

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

   if 1.75e71 < x.re

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. *-commutative 70.4%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 90.7%

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

      1. difference-of-squares 72.2%

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

      2. *-commutative 72.2%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re - 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.75 \cdot 10^{+71}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;t\_0 - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))
   (*
    x.re_s
    (if (<= x.re_m 1.4e+53) (- t_0 (* x.im (* (* x.re_m x.im) 2.0))) t_0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (x_46_re_m <= 1.4e+53) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    if (x_46re_m <= 1.4d+53) then
        tmp = t_0 - (x_46im * ((x_46re_m * x_46im) * 2.0d0))
    else
        tmp = t_0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (x_46_re_m <= 1.4e+53) {
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	tmp = 0
	if x_46_re_m <= 1.4e+53:
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = t_0
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)))
	tmp = 0.0
	if (x_46_re_m <= 1.4e+53)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	tmp = 0.0;
	if (x_46_re_m <= 1.4e+53)
		tmp = t_0 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.4e+53], N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.4e53

   1. Initial program 87.6%

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

      1. difference-of-squares 90.1%

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

      2. *-commutative 90.1%

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

   4. Applied egg-rr 90.1%

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

      1. *-un-lft-identity 90.1%

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

      2. *-commutative 90.1%

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

      3. *-un-lft-identity 90.1%

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

      4. distribute-rgt-out 90.1%

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

      5. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

   if 1.4e53 < x.re

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

      2. *-commutative 72.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 91.5%

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

      1. difference-of-squares 74.5%

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

      2. *-commutative 74.5%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re - 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+51}:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right) - x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+51)
    (-
     (* (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)))
     (* x.im (* x.re_m (* x.im 2.0))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+51) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1d+51) then
        tmp = ((x_46re_m - x_46im) * (x_46re_m * (x_46re_m + x_46im))) - (x_46im * (x_46re_m * (x_46im * 2.0d0)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+51) {
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1e+51:
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+51)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m + x_46_im))) - Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * 2.0))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1e+51)
		tmp = ((x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m + x_46_im))) - (x_46_im * (x_46_re_m * (x_46_im * 2.0)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e+51], N[(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] - N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1e51

   1. Initial program 87.6%

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

      1. difference-of-squares 90.1%

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

      2. *-commutative 90.1%

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

   4. Applied egg-rr 90.1%

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

      1. *-un-lft-identity 90.1%

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

      2. *-commutative 90.1%

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

      3. *-un-lft-identity 90.1%

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

      4. distribute-rgt-out 90.1%

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

      5. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. cancel-sign-sub-inv 90.1%

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

      2. associate-*l* 96.2%

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

      3. +-commutative 96.2%

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

      4. associate-*l* 96.2%

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

   8. Applied egg-rr 96.2%

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

   if 1e51 < x.re

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

      2. *-commutative 72.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 91.5%

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

      1. difference-of-squares 74.5%

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

      2. *-commutative 74.5%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re - 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+51}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 2.1× speedup

Math

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

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.re_m x.im) (+ 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_re_m - x_46_im) * (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_46re_m - x_46im) * (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_re_m - x_46_im) * (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_re_m - x_46_im) * (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(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * 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_re_m - x_46_im) * (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[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

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

   2. *-commutative 84.2%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

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

   5. metadata-eval 0.0%

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

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

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

   8. associate-*r/ 0.0%

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

   9. metadata-eval 0.0%

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

   10. metadata-eval 0.0%

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

4. Applied egg-rr 0.0%

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

6. Simplified 73.7%

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

   1. difference-of-squares 86.5%

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

   2. *-commutative 86.5%

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

8. Applied egg-rr 78.8%

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

9. Final simplification 78.8%

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

Developer target: 87.5% 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 2024089 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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