math.cube on complex, real part

Percentage Accurate: 82.6% → 98.8%

Time: 13.1s

Alternatives: 11

Speedup: 1.2×

• 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: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2.3e+153)
   (* (* x.re x.re) (+ x.re x.im))
   (if (<= x.re 2e+101)
     (fma (* x.re x.im) (* x.im -3.0) (pow x.re 3.0))
     (* (* x.re x.re) (- x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.3e+153) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else if (x_46_re <= 2e+101) {
		tmp = fma((x_46_re * x_46_im), (x_46_im * -3.0), pow(x_46_re, 3.0));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2.3e+153)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im));
	elseif (x_46_re <= 2e+101)
		tmp = fma(Float64(x_46_re * x_46_im), Float64(x_46_im * -3.0), (x_46_re ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2.3e+153], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2e+101], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.3000000000000001e153

   1. Initial program 55.0%

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

      1. sqr-neg 55.0%

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

      2. difference-of-squares 70.0%

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

      3. sub-neg 70.0%

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

      4. associate-*l* 70.0%

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

      5. sub-neg 70.0%

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

      6. remove-double-neg 70.0%

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

      7. +-commutative 70.0%

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

      8. *-commutative 70.0%

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

      9. *-commutative 70.0%

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

      10. distribute-rgt-out 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in x.re around inf 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

      1. expm1-log1p-u 7.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 7.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 7.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 7.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 15.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 15.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 7.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 7.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 7.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 7.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 62.5%

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

   10. Simplified 62.5%

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

   11. Taylor expanded in x.im around 0 30.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 30.0%

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

       2. cube-mult 30.0%

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

       3. distribute-rgt-in 92.5%

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

       4. +-commutative 92.5%

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

   13. Simplified 92.5%

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

   if -2.3000000000000001e153 < x.re < 2e101

   1. Initial program 92.0%

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

   3. Simplified 92.0%

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

      1. associate-*r* 92.1%

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

      2. associate-*l* 92.1%

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

      3. +-commutative 92.1%

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

      4. associate-*l* 92.1%

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

      5. associate-*r* 92.0%

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

      6. associate-*r* 99.2%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 2e101 < x.re

   1. Initial program 64.4%

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

      1. sqr-neg 64.4%

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

      2. difference-of-squares 77.8%

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

      3. sub-neg 77.8%

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

      4. associate-*l* 77.8%

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

      5. sub-neg 77.8%

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

      6. remove-double-neg 77.8%

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

      7. +-commutative 77.8%

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

      8. *-commutative 77.8%

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

      9. *-commutative 77.8%

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

      10. distribute-rgt-out 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in x.re around inf 77.8%

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

      1. unpow2 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in x.re around inf 46.7%

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

      1. +-commutative 46.7%

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

      2. cube-mult 46.7%

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

      3. unpow2 46.7%

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

      4. associate-*r* 46.7%

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

      5. distribute-rgt-out 100.0%

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

      6. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.8 \cdot 10^{+130}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+101}:\\ \;\;\;\;{x.re}^{3} + \left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.8e+130)
   (* (* x.re x.re) (+ x.re x.im))
   (if (<= x.re 2e+101)
     (+ (pow x.re 3.0) (* (* x.im -3.0) (* x.re x.im)))
     (* (* x.re x.re) (- x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.8e+130) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else if (x_46_re <= 2e+101) {
		tmp = pow(x_46_re, 3.0) + ((x_46_im * -3.0) * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.8d+130)) then
        tmp = (x_46re * x_46re) * (x_46re + x_46im)
    else if (x_46re <= 2d+101) then
        tmp = (x_46re ** 3.0d0) + ((x_46im * (-3.0d0)) * (x_46re * x_46im))
    else
        tmp = (x_46re * x_46re) * (x_46re - x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.8e+130) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else if (x_46_re <= 2e+101) {
		tmp = Math.pow(x_46_re, 3.0) + ((x_46_im * -3.0) * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.8e+130:
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im)
	elif x_46_re <= 2e+101:
		tmp = math.pow(x_46_re, 3.0) + ((x_46_im * -3.0) * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.8e+130)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im));
	elseif (x_46_re <= 2e+101)
		tmp = Float64((x_46_re ^ 3.0) + Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.8e+130)
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	elseif (x_46_re <= 2e+101)
		tmp = (x_46_re ^ 3.0) + ((x_46_im * -3.0) * (x_46_re * x_46_im));
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.8e+130], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2e+101], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.8000000000000001e130

   1. Initial program 54.8%

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

      1. sqr-neg 54.8%

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

      2. difference-of-squares 69.0%

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

      3. sub-neg 69.0%

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

      4. associate-*l* 69.0%

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

      5. sub-neg 69.0%

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

      6. remove-double-neg 69.0%

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

      7. +-commutative 69.0%

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

      8. *-commutative 69.0%

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

      9. *-commutative 69.0%

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

      10. distribute-rgt-out 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in x.re around inf 61.9%

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

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

      1. expm1-log1p-u 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in x.im around 0 31.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 31.0%

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

       2. cube-mult 31.0%

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

       3. distribute-rgt-in 92.9%

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

       4. +-commutative 92.9%

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

   13. Simplified 92.9%

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

   if -1.8000000000000001e130 < x.re < 2e101

   1. Initial program 92.5%

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

   3. Simplified 92.5%

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

      1. associate-*r* 92.6%

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

      2. associate-*l* 92.6%

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

      3. +-commutative 92.6%

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

      4. associate-*l* 92.6%

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

      5. associate-*r* 92.5%

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

      6. associate-*r* 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 2e101 < x.re

   1. Initial program 64.4%

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

      1. sqr-neg 64.4%

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

      2. difference-of-squares 77.8%

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

      3. sub-neg 77.8%

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

      4. associate-*l* 77.8%

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

      5. sub-neg 77.8%

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

      6. remove-double-neg 77.8%

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

      7. +-commutative 77.8%

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

      8. *-commutative 77.8%

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

      9. *-commutative 77.8%

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

      10. distribute-rgt-out 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in x.re around inf 77.8%

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

      1. unpow2 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in x.re around inf 46.7%

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

      1. +-commutative 46.7%

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

      2. cube-mult 46.7%

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

      3. unpow2 46.7%

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

      4. associate-*r* 46.7%

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

      5. distribute-rgt-out 100.0%

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

      6. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.8 \cdot 10^{+130}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+101}:\\ \;\;\;\;{x.re}^{3} + \left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1e+140)
   (* (* x.re x.re) (+ x.re x.im))
   (if (<= x.re 5.1e+73)
     (-
      (* (- x.re x.im) (* x.re (+ x.re x.im)))
      (* x.im (* x.re (+ x.im x.im))))
     (* (* x.re x.re) (- x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else if (x_46_re <= 5.1e+73) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1d+140)) then
        tmp = (x_46re * x_46re) * (x_46re + x_46im)
    else if (x_46re <= 5.1d+73) then
        tmp = ((x_46re - x_46im) * (x_46re * (x_46re + x_46im))) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = (x_46re * x_46re) * (x_46re - x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else if (x_46_re <= 5.1e+73) {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1e+140:
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im)
	elif x_46_re <= 5.1e+73:
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1e+140)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im));
	elseif (x_46_re <= 5.1e+73)
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re * Float64(x_46_re + x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1e+140)
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	elseif (x_46_re <= 5.1e+73)
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1e+140], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.1e+73], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.00000000000000006e140

   1. Initial program 54.8%

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

      1. sqr-neg 54.8%

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

      2. difference-of-squares 69.0%

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

      3. sub-neg 69.0%

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

      4. associate-*l* 69.0%

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

      5. sub-neg 69.0%

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

      6. remove-double-neg 69.0%

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

      7. +-commutative 69.0%

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

      8. *-commutative 69.0%

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

      9. *-commutative 69.0%

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

      10. distribute-rgt-out 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in x.re around inf 61.9%

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

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

      1. expm1-log1p-u 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in x.im around 0 31.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 31.0%

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

       2. cube-mult 31.0%

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

       3. distribute-rgt-in 92.9%

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

       4. +-commutative 92.9%

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

   13. Simplified 92.9%

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

   if -1.00000000000000006e140 < x.re < 5.10000000000000024e73

   1. Initial program 92.1%

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

      1. sqr-neg 92.1%

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

      2. difference-of-squares 92.1%

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

      3. sub-neg 92.1%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   if 5.10000000000000024e73 < x.re

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. difference-of-squares 81.4%

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

      3. sub-neg 81.4%

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

      4. associate-*l* 81.4%

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

      5. sub-neg 81.4%

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

      6. remove-double-neg 81.4%

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

      7. +-commutative 81.4%

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

      8. *-commutative 81.4%

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

      9. *-commutative 81.4%

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

      10. distribute-rgt-out 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in x.re around inf 77.7%

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

      1. unpow2 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x.re around inf 51.9%

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

      1. +-commutative 51.9%

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

      2. cube-mult 51.8%

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

      3. unpow2 51.8%

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

      4. associate-*r* 51.8%

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

      5. distribute-rgt-out 96.2%

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

      6. mul-1-neg 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -1.16 \cdot 10^{-63}:\\ \;\;\;\;t_0 - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* x.re x.re) (+ x.re x.im))))
   (if (<= x.re -1e+140)
     t_0
     (if (<= x.re -1.16e-63)
       (- t_0 (* x.im (* x.re (+ x.im x.im))))
       (if (<= x.re 3.9e-53)
         (* -3.0 (* x.im (* x.re x.im)))
         (* (* x.re x.re) (- x.re x.im)))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = t_0;
	} else if (x_46_re <= -1.16e-63) {
		tmp = t_0 - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else if (x_46_re <= 3.9e-53) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re * x_46re) * (x_46re + x_46im)
    if (x_46re <= (-1d+140)) then
        tmp = t_0
    else if (x_46re <= (-1.16d-63)) then
        tmp = t_0 - (x_46im * (x_46re * (x_46im + x_46im)))
    else if (x_46re <= 3.9d-53) then
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    else
        tmp = (x_46re * x_46re) * (x_46re - x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = t_0;
	} else if (x_46_re <= -1.16e-63) {
		tmp = t_0 - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else if (x_46_re <= 3.9e-53) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im)
	tmp = 0
	if x_46_re <= -1e+140:
		tmp = t_0
	elif x_46_re <= -1.16e-63:
		tmp = t_0 - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	elif x_46_re <= 3.9e-53:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im))
	tmp = 0.0
	if (x_46_re <= -1e+140)
		tmp = t_0;
	elseif (x_46_re <= -1.16e-63)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	elseif (x_46_re <= 3.9e-53)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	tmp = 0.0;
	if (x_46_re <= -1e+140)
		tmp = t_0;
	elseif (x_46_re <= -1.16e-63)
		tmp = t_0 - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	elseif (x_46_re <= 3.9e-53)
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1e+140], t$95$0, If[LessEqual[x$46$re, -1.16e-63], N[(t$95$0 - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 3.9e-53], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.00000000000000006e140

   1. Initial program 54.8%

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

      1. sqr-neg 54.8%

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

      2. difference-of-squares 69.0%

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

      3. sub-neg 69.0%

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

      4. associate-*l* 69.0%

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

      5. sub-neg 69.0%

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

      6. remove-double-neg 69.0%

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

      7. +-commutative 69.0%

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

      8. *-commutative 69.0%

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

      9. *-commutative 69.0%

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

      10. distribute-rgt-out 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in x.re around inf 61.9%

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

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

      1. expm1-log1p-u 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 14.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 7.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in x.im around 0 31.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 31.0%

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

       2. cube-mult 31.0%

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

       3. distribute-rgt-in 92.9%

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

       4. +-commutative 92.9%

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

   13. Simplified 92.9%

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

   if -1.00000000000000006e140 < x.re < -1.16e-63

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. difference-of-squares 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*l* 99.6%

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

      5. sub-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. distribute-rgt-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x.re around inf 86.2%

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

      1. unpow2 86.2%

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

   6. Simplified 86.2%

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

      1. expm1-log1p-u 28.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 9.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 9.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 9.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 18.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 18.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 8.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 10.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 10.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 10.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 29.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 89.1%

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

   10. Simplified 89.1%

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

   if -1.16e-63 < x.re < 3.9000000000000002e-53

   1. Initial program 88.6%

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

      1. sqr-neg 88.6%

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

      2. difference-of-squares 88.6%

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

      3. sub-neg 88.6%

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

      4. associate-*l* 99.6%

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

      5. sub-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. distribute-rgt-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x.re around 0 84.1%

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

      1. distribute-rgt-out-- 84.1%

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

      2. metadata-eval 84.1%

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

      3. associate-*r* 84.1%

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

      4. *-commutative 84.1%

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

      5. associate-*r* 84.0%

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

      6. unpow2 84.0%

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

      7. metadata-eval 84.0%

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

      8. distribute-rgt-out-- 84.0%

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

      9. associate-*l* 95.1%

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

      10. distribute-rgt-out-- 95.1%

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

      11. metadata-eval 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in x.im around 0 84.1%

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

      1. unpow2 84.1%

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

      2. associate-*r* 95.2%

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

   9. Simplified 95.2%

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

   if 3.9000000000000002e-53 < x.re

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. difference-of-squares 85.8%

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

      3. sub-neg 85.8%

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

      4. associate-*l* 85.8%

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

      5. sub-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. +-commutative 85.8%

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

      8. *-commutative 85.8%

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

      9. *-commutative 85.8%

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

      10. distribute-rgt-out 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in x.re around inf 78.5%

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

      1. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in x.re around inf 57.9%

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

      1. +-commutative 57.9%

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

      2. cube-mult 57.8%

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

      3. unpow2 57.8%

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

      4. associate-*r* 57.8%

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

      5. distribute-rgt-out 91.6%

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

      6. mul-1-neg 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq -1.16 \cdot 10^{-63}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 5: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -23000 \lor \neg \left(x.re \leq -6.8 \cdot 10^{-25}\right) \land \left(x.re \leq -2.15 \cdot 10^{-60} \lor \neg \left(x.re \leq 3.1 \cdot 10^{-52}\right)\right):\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -23000.0)
         (and (not (<= x.re -6.8e-25))
              (or (<= x.re -2.15e-60) (not (<= x.re 3.1e-52)))))
   (* (* x.re x.re) (+ x.re x.im))
   (* -3.0 (* x.im (* x.re x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -23000.0) || (!(x_46_re <= -6.8e-25) && ((x_46_re <= -2.15e-60) || !(x_46_re <= 3.1e-52)))) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-23000.0d0)) .or. (.not. (x_46re <= (-6.8d-25))) .and. (x_46re <= (-2.15d-60)) .or. (.not. (x_46re <= 3.1d-52))) then
        tmp = (x_46re * x_46re) * (x_46re + x_46im)
    else
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -23000.0) || (!(x_46_re <= -6.8e-25) && ((x_46_re <= -2.15e-60) || !(x_46_re <= 3.1e-52)))) {
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -23000.0) or (not (x_46_re <= -6.8e-25) and ((x_46_re <= -2.15e-60) or not (x_46_re <= 3.1e-52))):
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im)
	else:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -23000.0) || (!(x_46_re <= -6.8e-25) && ((x_46_re <= -2.15e-60) || !(x_46_re <= 3.1e-52))))
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -23000.0) || (~((x_46_re <= -6.8e-25)) && ((x_46_re <= -2.15e-60) || ~((x_46_re <= 3.1e-52)))))
		tmp = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	else
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -23000.0], And[N[Not[LessEqual[x$46$re, -6.8e-25]], $MachinePrecision], Or[LessEqual[x$46$re, -2.15e-60], N[Not[LessEqual[x$46$re, 3.1e-52]], $MachinePrecision]]]], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -23000 or -6.80000000000000003e-25 < x.re < -2.15e-60 or 3.0999999999999999e-52 < x.re

   1. Initial program 75.2%

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

      1. sqr-neg 75.2%

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

      2. difference-of-squares 83.6%

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

      3. sub-neg 83.6%

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

      4. associate-*l* 83.6%

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

      5. sub-neg 83.6%

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

      6. remove-double-neg 83.6%

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

      7. +-commutative 83.6%

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

      8. *-commutative 83.6%

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

      9. *-commutative 83.6%

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

      10. distribute-rgt-out 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in x.re around inf 75.2%

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

      1. unpow2 75.2%

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

   6. Simplified 75.2%

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

      1. expm1-log1p-u 39.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 31.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 31.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 16.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 34.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 34.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 17.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 32.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 32.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 40.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 71.7%

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

   10. Simplified 71.7%

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

   11. Taylor expanded in x.im around 0 56.2%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 56.2%

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

       2. cube-mult 56.1%

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

       3. distribute-rgt-in 87.1%

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

       4. +-commutative 87.1%

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

   13. Simplified 87.1%

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

   if -23000 < x.re < -6.80000000000000003e-25 or -2.15e-60 < x.re < 3.0999999999999999e-52

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. difference-of-squares 89.1%

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

      3. sub-neg 89.1%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 84.8%

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

      1. distribute-rgt-out-- 84.8%

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

      2. metadata-eval 84.8%

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

      3. associate-*r* 84.8%

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

      4. *-commutative 84.8%

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

      5. associate-*r* 84.7%

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

      6. unpow2 84.7%

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

      7. metadata-eval 84.7%

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

      8. distribute-rgt-out-- 84.7%

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

      9. associate-*l* 95.4%

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

      10. distribute-rgt-out-- 95.4%

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

      11. metadata-eval 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in x.im around 0 84.8%

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

      1. unpow2 84.8%

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

      2. associate-*r* 95.4%

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

   9. Simplified 95.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -23000 \lor \neg \left(x.re \leq -6.8 \cdot 10^{-25}\right) \land \left(x.re \leq -2.15 \cdot 10^{-60} \lor \neg \left(x.re \leq 3.1 \cdot 10^{-52}\right)\right):\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 6: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ t_1 := -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.re \leq -26500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -7.5 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* x.re x.re) (+ x.re x.im)))
        (t_1 (* -3.0 (* x.im (* x.re x.im)))))
   (if (<= x.re -26500.0)
     t_0
     (if (<= x.re -4e-25)
       t_1
       (if (<= x.re -7.5e-64)
         t_0
         (if (<= x.re 5.5e-52) t_1 (* (* x.re x.re) (- x.re x.im))))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	double t_1 = -3.0 * (x_46_im * (x_46_re * x_46_im));
	double tmp;
	if (x_46_re <= -26500.0) {
		tmp = t_0;
	} else if (x_46_re <= -4e-25) {
		tmp = t_1;
	} else if (x_46_re <= -7.5e-64) {
		tmp = t_0;
	} else if (x_46_re <= 5.5e-52) {
		tmp = t_1;
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re * x_46re) * (x_46re + x_46im)
    t_1 = (-3.0d0) * (x_46im * (x_46re * x_46im))
    if (x_46re <= (-26500.0d0)) then
        tmp = t_0
    else if (x_46re <= (-4d-25)) then
        tmp = t_1
    else if (x_46re <= (-7.5d-64)) then
        tmp = t_0
    else if (x_46re <= 5.5d-52) then
        tmp = t_1
    else
        tmp = (x_46re * x_46re) * (x_46re - x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	double t_1 = -3.0 * (x_46_im * (x_46_re * x_46_im));
	double tmp;
	if (x_46_re <= -26500.0) {
		tmp = t_0;
	} else if (x_46_re <= -4e-25) {
		tmp = t_1;
	} else if (x_46_re <= -7.5e-64) {
		tmp = t_0;
	} else if (x_46_re <= 5.5e-52) {
		tmp = t_1;
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im)
	t_1 = -3.0 * (x_46_im * (x_46_re * x_46_im))
	tmp = 0
	if x_46_re <= -26500.0:
		tmp = t_0
	elif x_46_re <= -4e-25:
		tmp = t_1
	elif x_46_re <= -7.5e-64:
		tmp = t_0
	elif x_46_re <= 5.5e-52:
		tmp = t_1
	else:
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re + x_46_im))
	t_1 = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)))
	tmp = 0.0
	if (x_46_re <= -26500.0)
		tmp = t_0;
	elseif (x_46_re <= -4e-25)
		tmp = t_1;
	elseif (x_46_re <= -7.5e-64)
		tmp = t_0;
	elseif (x_46_re <= 5.5e-52)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re * x_46_re) * (x_46_re + x_46_im);
	t_1 = -3.0 * (x_46_im * (x_46_re * x_46_im));
	tmp = 0.0;
	if (x_46_re <= -26500.0)
		tmp = t_0;
	elseif (x_46_re <= -4e-25)
		tmp = t_1;
	elseif (x_46_re <= -7.5e-64)
		tmp = t_0;
	elseif (x_46_re <= 5.5e-52)
		tmp = t_1;
	else
		tmp = (x_46_re * x_46_re) * (x_46_re - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -26500.0], t$95$0, If[LessEqual[x$46$re, -4e-25], t$95$1, If[LessEqual[x$46$re, -7.5e-64], t$95$0, If[LessEqual[x$46$re, 5.5e-52], t$95$1, N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -26500 or -4.00000000000000015e-25 < x.re < -7.49999999999999949e-64

   1. Initial program 73.0%

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

      1. sqr-neg 73.0%

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

      2. difference-of-squares 81.5%

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

      3. sub-neg 81.5%

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

      4. associate-*l* 81.5%

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

      5. sub-neg 81.5%

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

      6. remove-double-neg 81.5%

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

      7. +-commutative 81.5%

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

      8. *-commutative 81.5%

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

      9. *-commutative 81.5%

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

      10. distribute-rgt-out 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x.re around inf 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

      1. expm1-log1p-u 14.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 5.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 5.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 5.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 11.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 11.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 5.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 6.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 6.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 15.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 73.4%

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

   10. Simplified 73.4%

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

   11. Taylor expanded in x.im around 0 54.4%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 54.4%

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

       2. cube-mult 54.3%

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

       3. distribute-rgt-in 90.9%

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

       4. +-commutative 90.9%

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

   13. Simplified 90.9%

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

   if -26500 < x.re < -4.00000000000000015e-25 or -7.49999999999999949e-64 < x.re < 5.5e-52

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. difference-of-squares 89.1%

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

      3. sub-neg 89.1%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 84.8%

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

      1. distribute-rgt-out-- 84.8%

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

      2. metadata-eval 84.8%

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

      3. associate-*r* 84.8%

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

      4. *-commutative 84.8%

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

      5. associate-*r* 84.7%

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

      6. unpow2 84.7%

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

      7. metadata-eval 84.7%

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

      8. distribute-rgt-out-- 84.7%

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

      9. associate-*l* 95.4%

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

      10. distribute-rgt-out-- 95.4%

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

      11. metadata-eval 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in x.im around 0 84.8%

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

      1. unpow2 84.8%

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

      2. associate-*r* 95.4%

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

   9. Simplified 95.4%

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

   if 5.5e-52 < x.re

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. difference-of-squares 85.8%

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

      3. sub-neg 85.8%

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

      4. associate-*l* 85.8%

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

      5. sub-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. +-commutative 85.8%

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

      8. *-commutative 85.8%

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

      9. *-commutative 85.8%

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

      10. distribute-rgt-out 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in x.re around inf 78.5%

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

      1. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in x.re around inf 57.9%

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

      1. +-commutative 57.9%

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

      2. cube-mult 57.8%

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

      3. unpow2 57.8%

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

      4. associate-*r* 57.8%

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

      5. distribute-rgt-out 91.6%

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

      6. mul-1-neg 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -26500:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-25}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -7.5 \cdot 10^{-64}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.re \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 7: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.22 \cdot 10^{+96}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -4.3e+186)
   (* x.re (* x.im x.im))
   (if (<= x.re 1.22e+96)
     (* -3.0 (* x.im (* x.re x.im)))
     (* (* x.re x.re) x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -4.3e+186) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else if (x_46_re <= 1.22e+96) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-4.3d+186)) then
        tmp = x_46re * (x_46im * x_46im)
    else if (x_46re <= 1.22d+96) then
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -4.3e+186) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else if (x_46_re <= 1.22e+96) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -4.3e+186:
		tmp = x_46_re * (x_46_im * x_46_im)
	elif x_46_re <= 1.22e+96:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -4.3e+186)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	elseif (x_46_re <= 1.22e+96)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -4.3e+186)
		tmp = x_46_re * (x_46_im * x_46_im);
	elseif (x_46_re <= 1.22e+96)
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -4.3e+186], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.22e+96], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -4.3e186

   1. Initial program 55.6%

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

      1. sqr-neg 55.6%

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

      2. difference-of-squares 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-*l* 66.7%

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

      5. sub-neg 66.7%

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

      6. remove-double-neg 66.7%

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

      7. +-commutative 66.7%

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

      8. *-commutative 66.7%

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

      9. *-commutative 66.7%

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

      10. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

   5. Applied egg-rr 66.7%

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

   6. Taylor expanded in x.re around 0 11.6%

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

      1. mul-1-neg 11.6%

         \[\leadsto \color{blue}{-{x.im}^{2} \cdot x.re} \]

      2. unpow2 11.6%

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

      3. associate-*r* 11.5%

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

      4. distribute-rgt-neg-in 11.5%

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

      5. distribute-rgt-neg-in 11.5%

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

   8. Simplified 11.5%

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

      1. expm1-log1p-u 11.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

      2. expm1-udef 11.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

      3. *-commutative 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot \left(-x.re\right)\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-x.re\right) \cdot x.im\right)} \cdot x.im\right)} - 1 \]

      5. associate-*l* 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-x.re\right) \cdot \left(x.im \cdot x.im\right)}\right)} - 1 \]

      6. add-sqr-sqrt 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      7. sqrt-unprod 11.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      8. sqr-neg 11.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x.re \cdot x.re}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      9. sqrt-prod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      10. add-sqr-sqrt 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.re} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      11. associate-*r* 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      12. *-commutative 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot x.im\right)} - 1 \]

      13. associate-*l* 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(x.re \cdot x.im\right)}\right)} - 1 \]

      14. *-commutative 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)} - 1 \]

   10. Applied egg-rr 0.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 0.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right)} \]

       2. expm1-log1p 35.5%

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

       3. associate-*r* 35.4%

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

       4. *-commutative 35.4%

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

   12. Simplified 35.4%

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

   if -4.3e186 < x.re < 1.21999999999999992e96

   1. Initial program 90.9%

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

      1. sqr-neg 90.9%

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

      2. difference-of-squares 92.1%

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

      3. sub-neg 92.1%

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

      4. associate-*l* 99.1%

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

      5. sub-neg 99.1%

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

      6. remove-double-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. *-commutative 99.1%

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

      10. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x.re around 0 64.9%

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

      1. distribute-rgt-out-- 64.9%

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

      2. metadata-eval 64.9%

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

      3. associate-*r* 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-*r* 64.9%

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

      6. unpow2 64.9%

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

      7. metadata-eval 64.9%

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

      8. distribute-rgt-out-- 64.9%

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

      9. associate-*l* 71.9%

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

      10. distribute-rgt-out-- 71.9%

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

      11. metadata-eval 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in x.im around 0 64.9%

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

      1. unpow2 64.9%

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

      2. associate-*r* 71.9%

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

   9. Simplified 71.9%

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

   if 1.21999999999999992e96 < x.re

   1. Initial program 66.6%

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

      1. sqr-neg 66.6%

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

      2. difference-of-squares 79.1%

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

      3. sub-neg 79.1%

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

      4. associate-*l* 79.1%

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

      5. sub-neg 79.1%

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

      6. remove-double-neg 79.1%

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

      7. +-commutative 79.1%

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

      8. *-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt-out 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in x.re around inf 79.1%

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

      1. unpow2 79.1%

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

   6. Simplified 79.1%

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

      1. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 66.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 35.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 30.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 66.6%

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

   10. Simplified 66.6%

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

   11. Taylor expanded in x.im around 0 50.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 50.0%

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

       2. cube-mult 49.9%

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

       3. distribute-rgt-in 87.4%

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

       4. +-commutative 87.4%

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

   13. Simplified 87.4%

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

   14. Taylor expanded in x.re around 0 37.7%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
   15. Step-by-step derivation

       1. unpow2 37.7%

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

   16. Simplified 37.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.22 \cdot 10^{+96}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 8: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.12 \cdot 10^{+187}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.22 \cdot 10^{+96}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.12e+187)
   (* x.re (* 3.0 (* x.im x.im)))
   (if (<= x.re 1.22e+96)
     (* -3.0 (* x.im (* x.re x.im)))
     (* (* x.re x.re) x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.12e+187) {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_im));
	} else if (x_46_re <= 1.22e+96) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.12d+187)) then
        tmp = x_46re * (3.0d0 * (x_46im * x_46im))
    else if (x_46re <= 1.22d+96) then
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.12e+187) {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_im));
	} else if (x_46_re <= 1.22e+96) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.12e+187:
		tmp = x_46_re * (3.0 * (x_46_im * x_46_im))
	elif x_46_re <= 1.22e+96:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.12e+187)
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_im)));
	elseif (x_46_re <= 1.22e+96)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.12e+187)
		tmp = x_46_re * (3.0 * (x_46_im * x_46_im));
	elseif (x_46_re <= 1.22e+96)
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.12e+187], N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.22e+96], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.12000000000000007e187

   1. Initial program 55.6%

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

      1. sqr-neg 55.6%

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

      2. difference-of-squares 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-*l* 66.7%

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

      5. sub-neg 66.7%

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

      6. remove-double-neg 66.7%

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

      7. +-commutative 66.7%

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

      8. *-commutative 66.7%

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

      9. *-commutative 66.7%

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

      10. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x.re around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. neg-mul-1 11.5%

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

      3. unpow2 11.5%

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

      4. distribute-rgt-neg-in 11.5%

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

   6. Simplified 11.5%

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

      1. cancel-sign-sub-inv 11.5%

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

      2. associate-*l* 11.5%

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

      3. add-sqr-sqrt 5.7%

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

      4. sqrt-unprod 6.0%

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

      5. sqr-neg 6.0%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 0.4%

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

      8. add-sqr-sqrt 0.2%

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

      9. sqrt-unprod 26.7%

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

      10. sqr-neg 26.7%

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

      11. sqrt-unprod 26.4%

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

      12. add-sqr-sqrt 35.6%

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

      13. associate-*r* 35.6%

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

   8. Applied egg-rr 35.6%

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

      1. associate-*r* 35.6%

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

      2. *-commutative 35.6%

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

      3. *-commutative 35.6%

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

      4. associate-*l* 35.6%

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

      5. *-commutative 35.6%

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

      6. count-2 35.6%

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

      7. associate-*r* 35.6%

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

      8. metadata-eval 35.6%

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

      9. distribute-lft-in 35.6%

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

      10. metadata-eval 35.6%

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

      11. distribute-rgt1-in 35.6%

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

      12. metadata-eval 35.6%

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

   10. Simplified 35.6%

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

   if -1.12000000000000007e187 < x.re < 1.21999999999999992e96

   1. Initial program 90.9%

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

      1. sqr-neg 90.9%

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

      2. difference-of-squares 92.1%

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

      3. sub-neg 92.1%

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

      4. associate-*l* 99.1%

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

      5. sub-neg 99.1%

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

      6. remove-double-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. *-commutative 99.1%

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

      10. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x.re around 0 64.9%

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

      1. distribute-rgt-out-- 64.9%

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

      2. metadata-eval 64.9%

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

      3. associate-*r* 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-*r* 64.9%

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

      6. unpow2 64.9%

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

      7. metadata-eval 64.9%

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

      8. distribute-rgt-out-- 64.9%

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

      9. associate-*l* 71.9%

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

      10. distribute-rgt-out-- 71.9%

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

      11. metadata-eval 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in x.im around 0 64.9%

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

      1. unpow2 64.9%

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

      2. associate-*r* 71.9%

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

   9. Simplified 71.9%

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

   if 1.21999999999999992e96 < x.re

   1. Initial program 66.6%

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

      1. sqr-neg 66.6%

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

      2. difference-of-squares 79.1%

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

      3. sub-neg 79.1%

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

      4. associate-*l* 79.1%

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

      5. sub-neg 79.1%

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

      6. remove-double-neg 79.1%

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

      7. +-commutative 79.1%

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

      8. *-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. distribute-rgt-out 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in x.re around inf 79.1%

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

      1. unpow2 79.1%

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

   6. Simplified 79.1%

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

      1. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 66.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 35.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 30.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 66.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 66.6%

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

   10. Simplified 66.6%

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

   11. Taylor expanded in x.im around 0 50.0%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 50.0%

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

       2. cube-mult 49.9%

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

       3. distribute-rgt-in 87.4%

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

       4. +-commutative 87.4%

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

   13. Simplified 87.4%

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

   14. Taylor expanded in x.re around 0 37.7%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
   15. Step-by-step derivation

       1. unpow2 37.7%

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

   16. Simplified 37.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.12 \cdot 10^{+187}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.22 \cdot 10^{+96}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 9: 43.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 5.2e+45)
   (* (* x.re x.re) x.im)
   (if (<= x.im 5.5e+152) (* x.re (* x.im x.im)) (* x.im (* x.re (- x.im))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 5.2e+45) {
		tmp = (x_46_re * x_46_re) * x_46_im;
	} else if (x_46_im <= 5.5e+152) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 5.2d+45) then
        tmp = (x_46re * x_46re) * x_46im
    else if (x_46im <= 5.5d+152) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46im * (x_46re * -x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 5.2e+45) {
		tmp = (x_46_re * x_46_re) * x_46_im;
	} else if (x_46_im <= 5.5e+152) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 5.2e+45:
		tmp = (x_46_re * x_46_re) * x_46_im
	elif x_46_im <= 5.5e+152:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_im * (x_46_re * -x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 5.2e+45)
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	elseif (x_46_im <= 5.5e+152)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 5.2e+45)
		tmp = (x_46_re * x_46_re) * x_46_im;
	elseif (x_46_im <= 5.5e+152)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_im * (x_46_re * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 5.2e+45], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[x$46$im, 5.5e+152], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 5.20000000000000014e45

   1. Initial program 90.9%

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

      1. sqr-neg 90.9%

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

      2. difference-of-squares 92.4%

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

      3. sub-neg 92.4%

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

      4. associate-*l* 94.7%

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

      5. sub-neg 94.7%

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

      6. remove-double-neg 94.7%

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

      7. +-commutative 94.7%

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

      8. *-commutative 94.7%

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

      9. *-commutative 94.7%

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

      10. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x.re around inf 77.2%

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

      1. unpow2 77.2%

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

   6. Simplified 77.2%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 48.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 48.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 24.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 46.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 46.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 24.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 43.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 43.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 51.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in x.im around 0 58.9%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 58.9%

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

       2. cube-mult 58.8%

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

       3. distribute-rgt-in 74.0%

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

       4. +-commutative 74.0%

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

   13. Simplified 74.0%

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

   14. Taylor expanded in x.re around 0 34.2%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
   15. Step-by-step derivation

       1. unpow2 34.2%

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

   16. Simplified 34.2%

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

   if 5.20000000000000014e45 < x.im < 5.4999999999999999e152

   1. Initial program 57.0%

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

      1. sqr-neg 57.0%

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

      2. difference-of-squares 57.0%

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

      3. sub-neg 57.0%

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

      4. associate-*l* 56.8%

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

      5. sub-neg 56.8%

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

      6. remove-double-neg 56.8%

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

      7. +-commutative 56.8%

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

      8. *-commutative 56.8%

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

      9. *-commutative 56.8%

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

      10. distribute-rgt-out 56.8%

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

   3. Simplified 56.8%

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

      1. add-cube-cbrt 56.4%

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

      2. pow3 56.5%

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

   5. Applied egg-rr 56.5%

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

   6. Taylor expanded in x.re around 0 8.9%

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

      1. mul-1-neg 8.9%

         \[\leadsto \color{blue}{-{x.im}^{2} \cdot x.re} \]

      2. unpow2 8.9%

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

      3. associate-*r* 8.9%

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

      4. distribute-rgt-neg-in 8.9%

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

      5. distribute-rgt-neg-in 8.9%

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

   8. Simplified 8.9%

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

      1. expm1-log1p-u 2.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

      2. expm1-udef 1.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

      3. *-commutative 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot \left(-x.re\right)\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-x.re\right) \cdot x.im\right)} \cdot x.im\right)} - 1 \]

      5. associate-*l* 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-x.re\right) \cdot \left(x.im \cdot x.im\right)}\right)} - 1 \]

      6. add-sqr-sqrt 1.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      7. sqrt-unprod 9.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      8. sqr-neg 9.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x.re \cdot x.re}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      9. sqrt-prod 7.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      10. add-sqr-sqrt 7.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.re} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      11. associate-*r* 7.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      12. *-commutative 7.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot x.im\right)} - 1 \]

      13. associate-*l* 7.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(x.re \cdot x.im\right)}\right)} - 1 \]

      14. *-commutative 7.7%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)} - 1 \]

   10. Applied egg-rr 7.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 7.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right)} \]

       2. expm1-log1p 44.4%

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

       3. associate-*r* 44.4%

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

       4. *-commutative 44.4%

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

   12. Simplified 44.4%

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

   if 5.4999999999999999e152 < x.im

   1. Initial program 41.5%

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

      1. sqr-neg 41.5%

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

      2. difference-of-squares 71.5%

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

      3. sub-neg 71.5%

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

      4. associate-*l* 96.5%

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

      5. sub-neg 96.5%

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

      6. remove-double-neg 96.5%

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

      7. +-commutative 96.5%

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

      8. *-commutative 96.5%

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

      9. *-commutative 96.5%

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

      10. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. add-cube-cbrt 96.4%

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

      2. pow3 96.4%

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

   5. Applied egg-rr 96.4%

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

   6. Taylor expanded in x.re around 0 71.5%

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

      1. mul-1-neg 71.5%

         \[\leadsto \color{blue}{-{x.im}^{2} \cdot x.re} \]

      2. unpow2 71.5%

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

      3. associate-*r* 74.7%

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

      4. distribute-rgt-neg-in 74.7%

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

      5. distribute-rgt-neg-in 74.7%

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

   8. Simplified 74.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 10: 33.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.8 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -3.8e+181) (* x.re (* x.im x.im)) (* (* x.re x.re) x.im)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.8e+181) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-3.8d+181)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.8e+181) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -3.8e+181:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -3.8e+181)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -3.8e+181)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -3.8e+181], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.8000000000000001e181

   1. Initial program 56.8%

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

      1. sqr-neg 56.8%

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

      2. difference-of-squares 67.6%

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

      3. sub-neg 67.6%

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

      4. associate-*l* 67.6%

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

      5. sub-neg 67.6%

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

      6. remove-double-neg 67.6%

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

      7. +-commutative 67.6%

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

      8. *-commutative 67.6%

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

      9. *-commutative 67.6%

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

      10. distribute-rgt-out 67.6%

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

   3. Simplified 67.6%

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

      1. add-cube-cbrt 67.6%

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

      2. pow3 67.6%

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

   5. Applied egg-rr 67.6%

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

   6. Taylor expanded in x.re around 0 11.3%

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

      1. mul-1-neg 11.3%

         \[\leadsto \color{blue}{-{x.im}^{2} \cdot x.re} \]

      2. unpow2 11.3%

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

      3. associate-*r* 11.3%

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

      4. distribute-rgt-neg-in 11.3%

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

      5. distribute-rgt-neg-in 11.3%

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

   8. Simplified 11.3%

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

      1. expm1-log1p-u 11.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

      2. expm1-udef 11.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

      3. *-commutative 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot \left(-x.re\right)\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-x.re\right) \cdot x.im\right)} \cdot x.im\right)} - 1 \]

      5. associate-*l* 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-x.re\right) \cdot \left(x.im \cdot x.im\right)}\right)} - 1 \]

      6. add-sqr-sqrt 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      7. sqrt-unprod 10.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      8. sqr-neg 10.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x.re \cdot x.re}} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      9. sqrt-prod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      10. add-sqr-sqrt 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.re} \cdot \left(x.im \cdot x.im\right)\right)} - 1 \]

      11. associate-*r* 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      12. *-commutative 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot x.im\right)} - 1 \]

      13. associate-*l* 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(x.re \cdot x.im\right)}\right)} - 1 \]

      14. *-commutative 0.6%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)} - 1 \]

   10. Applied egg-rr 0.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 0.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right)} \]

       2. expm1-log1p 34.6%

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

       3. associate-*r* 34.5%

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

       4. *-commutative 34.5%

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

   12. Simplified 34.5%

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

   if -3.8000000000000001e181 < x.re

   1. Initial program 85.6%

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

      1. sqr-neg 85.6%

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

      2. difference-of-squares 89.2%

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

      3. sub-neg 89.2%

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

      4. associate-*l* 94.7%

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

      5. sub-neg 94.7%

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

      6. remove-double-neg 94.7%

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

      7. +-commutative 94.7%

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

      8. *-commutative 94.7%

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

      9. *-commutative 94.7%

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

      10. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x.re around inf 69.9%

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

      1. unpow2 69.9%

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

   6. Simplified 69.9%

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

      1. expm1-log1p-u 53.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-udef 45.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      3. sub-neg 45.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      4. add-sqr-sqrt 21.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. sqrt-unprod 45.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      6. sqr-neg 45.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      7. sqrt-unprod 28.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      8. add-sqr-sqrt 46.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. +-commutative 46.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   9. Step-by-step derivation

      1. expm1-def 54.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      2. expm1-log1p 67.6%

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

   10. Simplified 67.6%

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

   11. Taylor expanded in x.im around 0 50.3%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
   12. Step-by-step derivation

       1. unpow2 50.3%

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

       2. cube-mult 50.2%

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

       3. distribute-rgt-in 59.8%

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

       4. +-commutative 59.8%

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

   13. Simplified 59.8%

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

   14. Taylor expanded in x.re around 0 28.1%

       \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
   15. Step-by-step derivation

       1. unpow2 28.1%

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

   16. Simplified 28.1%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.8 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 11: 31.2% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* (* x.re x.re) x.im))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_re) * x_46_im;
}

Fortran

NOTE: x.im should be positive before calling this function
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
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_re) * x_46_im;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return (x_46_re * x_46_re) * x_46_im

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * x_46_re) * x_46_im)
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = (x_46_re * x_46_re) * x_46_im;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. sqr-neg 81.4%

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

   2. difference-of-squares 86.1%

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

   3. sub-neg 86.1%

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

   4. associate-*l* 90.8%

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

   5. sub-neg 90.8%

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

   6. remove-double-neg 90.8%

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

   7. +-commutative 90.8%

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

   8. *-commutative 90.8%

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

   9. *-commutative 90.8%

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

   10. distribute-rgt-out 90.8%

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

3. Simplified 90.8%

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

4. Taylor expanded in x.re around inf 68.7%

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

   1. unpow2 68.7%

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

6. Simplified 68.7%

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

   1. expm1-log1p-u 46.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   2. expm1-udef 40.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re - x.im\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   3. sub-neg 40.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.re + \left(-x.im\right)\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   4. add-sqr-sqrt 18.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   5. sqrt-unprod 40.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   6. sqr-neg 40.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   7. sqrt-unprod 25.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   8. add-sqr-sqrt 40.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(x.re + \color{blue}{x.im}\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   9. +-commutative 40.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re \cdot x.re\right)\right)} - 1\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

8. Applied egg-rr 40.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)} - 1\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 47.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.im + x.re\right) \cdot \left(x.re \cdot x.re\right)\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   2. expm1-log1p 66.8%

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

10. Simplified 66.8%

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

11. Taylor expanded in x.im around 0 47.7%

    \[\leadsto \color{blue}{x.im \cdot {x.re}^{2} + {x.re}^{3}} \]
12. Step-by-step derivation

    1. unpow2 47.7%

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

    2. cube-mult 47.6%

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

    3. distribute-rgt-in 64.8%

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

    4. +-commutative 64.8%

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

13. Simplified 64.8%

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

14. Taylor expanded in x.re around 0 29.5%

    \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
15. Step-by-step derivation

    1. unpow2 29.5%

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

16. Simplified 29.5%

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

17. Final simplification 29.5%

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

Developer target: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023283 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* 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)))