math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 96.5%

Time: 7.3s

Alternatives: 11

Speedup: 1.4×

• Unsound rule application detected in e-graph. Results may not be sound.

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 7.7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot 3, -{x.im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -7.5e+154)
   (* x.re (* x.re (* x.im 3.0)))
   (if (<= x.re 7.7e+149)
     (fma (* x.re x.re) (* x.im 3.0) (- (pow x.im 3.0)))
     (* x.re (* 3.0 (* x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -7.5e+154) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_re <= 7.7e+149) {
		tmp = fma((x_46_re * x_46_re), (x_46_im * 3.0), -pow(x_46_im, 3.0));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -7.5e+154)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	elseif (x_46_re <= 7.7e+149)
		tmp = fma(Float64(x_46_re * x_46_re), Float64(x_46_im * 3.0), Float64(-(x_46_im ^ 3.0)));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -7.5e+154], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.7e+149], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im * 3.0), $MachinePrecision] + (-N[Power[x$46$im, 3.0], $MachinePrecision])), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.5000000000000004e154

   1. Initial program 48.6%

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

      1. add-log-exp 48.6%

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

      2. *-un-lft-identity 48.6%

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

      3. log-prod 48.6%

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

      4. metadata-eval 48.6%

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

      5. add-log-exp 48.6%

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

      6. *-commutative 48.6%

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

      7. difference-of-squares 69.6%

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

      8. associate-*r* 92.0%

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

   3. Applied egg-rr 92.0%

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

   4. Taylor expanded in x.im around 0 69.6%

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

      1. distribute-lft1-in 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-commutative 69.6%

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

      4. associate-*r* 69.6%

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

      5. metadata-eval 69.6%

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

      6. distribute-lft1-in 69.6%

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

      7. unpow2 69.6%

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

      8. associate-*l* 92.0%

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

      9. distribute-lft1-in 92.0%

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

      10. metadata-eval 92.0%

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

      11. *-commutative 92.0%

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

   6. Simplified 92.0%

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

   if -7.5000000000000004e154 < x.re < 7.69999999999999998e149

   1. Initial program 89.6%

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

   2. Taylor expanded in x.re around 0 89.6%

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

   4. Simplified 99.9%

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

   if 7.69999999999999998e149 < x.re

   1. Initial program 65.8%

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

      1. add-log-exp 61.8%

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

      2. *-un-lft-identity 61.8%

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

      3. log-prod 61.8%

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

      4. metadata-eval 61.8%

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

      5. add-log-exp 65.8%

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

      6. *-commutative 65.8%

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

      7. difference-of-squares 74.5%

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

      8. associate-*r* 86.9%

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

   3. Applied egg-rr 86.9%

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

   4. Taylor expanded in x.im around 0 74.5%

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

      1. distribute-lft1-in 74.5%

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

      2. metadata-eval 74.5%

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

      3. *-commutative 74.5%

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

      4. associate-*r* 74.5%

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

      5. metadata-eval 74.5%

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

      6. distribute-lft1-in 74.5%

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

      7. unpow2 74.5%

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

      8. associate-*l* 86.9%

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

      9. distribute-lft1-in 86.9%

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

      10. metadata-eval 86.9%

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

      11. *-commutative 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in x.re around 0 87.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 7.7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot 3, -{x.im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.re + x.re\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -7.5e+154)
   (* x.re (* x.re (* x.im 3.0)))
   (if (<= x.re 6.5e+149)
     (fma x.im (* x.re (+ x.re x.re)) (* x.im (- (* x.re x.re) (* x.im x.im))))
     (* x.re (* 3.0 (* x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -7.5e+154) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_re <= 6.5e+149) {
		tmp = fma(x_46_im, (x_46_re * (x_46_re + x_46_re)), (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -7.5e+154)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	elseif (x_46_re <= 6.5e+149)
		tmp = fma(x_46_im, Float64(x_46_re * Float64(x_46_re + x_46_re)), Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -7.5e+154], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.5e+149], N[(x$46$im * N[(x$46$re * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.5000000000000004e154

   1. Initial program 48.6%

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

      1. add-log-exp 48.6%

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

      2. *-un-lft-identity 48.6%

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

      3. log-prod 48.6%

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

      4. metadata-eval 48.6%

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

      5. add-log-exp 48.6%

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

      6. *-commutative 48.6%

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

      7. difference-of-squares 69.6%

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

      8. associate-*r* 92.0%

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

   3. Applied egg-rr 92.0%

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

   4. Taylor expanded in x.im around 0 69.6%

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

      1. distribute-lft1-in 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-commutative 69.6%

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

      4. associate-*r* 69.6%

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

      5. metadata-eval 69.6%

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

      6. distribute-lft1-in 69.6%

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

      7. unpow2 69.6%

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

      8. associate-*l* 92.0%

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

      9. distribute-lft1-in 92.0%

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

      10. metadata-eval 92.0%

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

      11. *-commutative 92.0%

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

   6. Simplified 92.0%

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

   if -7.5000000000000004e154 < x.re < 6.50000000000000015e149

   1. Initial program 89.6%

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

      1. add-cube-cbrt 89.0%

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

      2. pow3 89.0%

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

      3. *-commutative 89.0%

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

      4. difference-of-squares 89.0%

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

      5. associate-*r* 89.0%

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

   3. Applied egg-rr 89.0%

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

   4. Taylor expanded in x.re around 0 89.0%

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

   6. Simplified 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-*l* 89.0%

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

      3. fma-def 99.3%

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

      4. unpow3 99.3%

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

      5. add-cube-cbrt 99.8%

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

      6. associate-*l* 99.8%

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

      7. difference-of-squares 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 6.50000000000000015e149 < x.re

   1. Initial program 65.8%

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

      1. add-log-exp 61.8%

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

      2. *-un-lft-identity 61.8%

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

      3. log-prod 61.8%

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

      4. metadata-eval 61.8%

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

      5. add-log-exp 65.8%

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

      6. *-commutative 65.8%

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

      7. difference-of-squares 74.5%

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

      8. associate-*r* 86.9%

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

   3. Applied egg-rr 86.9%

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

   4. Taylor expanded in x.im around 0 74.5%

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

      1. distribute-lft1-in 74.5%

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

      2. metadata-eval 74.5%

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

      3. *-commutative 74.5%

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

      4. associate-*r* 74.5%

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

      5. metadata-eval 74.5%

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

      6. distribute-lft1-in 74.5%

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

      7. unpow2 74.5%

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

      8. associate-*l* 86.9%

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

      9. distribute-lft1-in 86.9%

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

      10. metadata-eval 86.9%

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

      11. *-commutative 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in x.re around 0 87.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.re + x.re\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re + x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (+ (* (- x.re x.im) (* x.im (+ x.re x.im))) (* (* x.re x.im) (+ x.re x.re)))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re + x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re + x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re + x_46_re))
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im * Float64(x_46_re + x_46_im))) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re + x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re + x_46_re));
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.7%

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

      1. add-cube-cbrt 94.1%

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

      2. pow3 94.2%

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

      3. *-commutative 94.2%

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

      4. difference-of-squares 94.2%

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

      5. associate-*r* 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x.re around 0 94.2%

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

   6. Simplified 99.3%

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

      1. unpow3 99.3%

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

      2. add-cube-cbrt 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 55.6%

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

   6. Taylor expanded in x.re around 0 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re + x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 4: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.7 \cdot 10^{+72} \lor \neg \left(x.re \leq -3.4 \cdot 10^{+23} \lor \neg \left(x.re \leq -5.6 \cdot 10^{-64}\right) \land x.re \leq 4.9 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.7e+72)
         (not
          (or (<= x.re -3.4e+23)
              (and (not (<= x.re -5.6e-64)) (<= x.re 4.9e+60)))))
   (* x.im (* 3.0 (* x.re x.re)))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.7e+72) || !((x_46_re <= -3.4e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4.9e+60)))) {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

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 <= (-1.7d+72)) .or. (.not. (x_46re <= (-3.4d+23)) .or. (.not. (x_46re <= (-5.6d-64))) .and. (x_46re <= 4.9d+60))) then
        tmp = x_46im * (3.0d0 * (x_46re * x_46re))
    else
        tmp = x_46im * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.7e+72) || !((x_46_re <= -3.4e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4.9e+60)))) {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.7e+72) or not ((x_46_re <= -3.4e+23) or (not (x_46_re <= -5.6e-64) and (x_46_re <= 4.9e+60))):
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re))
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.7e+72) || !((x_46_re <= -3.4e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4.9e+60))))
		tmp = Float64(x_46_im * Float64(3.0 * Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.7e+72) || ~(((x_46_re <= -3.4e+23) || (~((x_46_re <= -5.6e-64)) && (x_46_re <= 4.9e+60)))))
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.7e+72], N[Not[Or[LessEqual[x$46$re, -3.4e+23], And[N[Not[LessEqual[x$46$re, -5.6e-64]], $MachinePrecision], LessEqual[x$46$re, 4.9e+60]]]], $MachinePrecision]], N[(x$46$im * N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.6999999999999999e72 or -3.39999999999999992e23 < x.re < -5.60000000000000008e-64 or 4.9000000000000003e60 < x.re

   1. Initial program 67.6%

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

   2. Taylor expanded in x.im around 0 70.9%

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

   4. Simplified 70.9%

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

   if -1.6999999999999999e72 < x.re < -3.39999999999999992e23 or -5.60000000000000008e-64 < x.re < 4.9000000000000003e60

   1. Initial program 93.9%

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

      1. expm1-log1p-u 67.2%

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

      2. expm1-udef 48.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 92.5%

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

   6. Taylor expanded in x.re around 0 91.5%

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

   8. Simplified 91.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.7 \cdot 10^{+72} \lor \neg \left(x.re \leq -3.4 \cdot 10^{+23} \lor \neg \left(x.re \leq -5.6 \cdot 10^{-64}\right) \land x.re \leq 4.9 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8 \cdot 10^{+71}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq -7 \cdot 10^{+23} \lor \neg \left(x.re \leq -5.6 \cdot 10^{-64}\right) \land x.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -8e+71)
   (* x.im (* x.re (* x.re 3.0)))
   (if (or (<= x.re -7e+23) (and (not (<= x.re -5.6e-64)) (<= x.re 4e+59)))
     (* x.im (* x.im (- x.im)))
     (* x.im (* 3.0 (* x.re x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -8e+71) {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	} else if ((x_46_re <= -7e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4e+59))) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

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 <= (-8d+71)) then
        tmp = x_46im * (x_46re * (x_46re * 3.0d0))
    else if ((x_46re <= (-7d+23)) .or. (.not. (x_46re <= (-5.6d-64))) .and. (x_46re <= 4d+59)) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46im * (3.0d0 * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -8e+71) {
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	} else if ((x_46_re <= -7e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4e+59))) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -8e+71:
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0))
	elif (x_46_re <= -7e+23) or (not (x_46_re <= -5.6e-64) and (x_46_re <= 4e+59)):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -8e+71)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_re * 3.0)));
	elseif ((x_46_re <= -7e+23) || (!(x_46_re <= -5.6e-64) && (x_46_re <= 4e+59)))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(3.0 * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -8e+71)
		tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
	elseif ((x_46_re <= -7e+23) || (~((x_46_re <= -5.6e-64)) && (x_46_re <= 4e+59)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -8e+71], N[(x$46$im * N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, -7e+23], And[N[Not[LessEqual[x$46$re, -5.6e-64]], $MachinePrecision], LessEqual[x$46$re, 4e+59]]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8.0000000000000003e71

   1. Initial program 55.1%

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

      1. add-log-exp 37.5%

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

      2. *-un-lft-identity 37.5%

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

      3. log-prod 37.5%

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

      4. metadata-eval 37.5%

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

      5. add-log-exp 55.1%

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

      6. *-commutative 55.1%

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

      7. difference-of-squares 69.2%

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

      8. associate-*r* 84.1%

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

   3. Applied egg-rr 84.1%

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

   4. Taylor expanded in x.im around 0 69.1%

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

      1. distribute-lft1-in 69.1%

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

      2. metadata-eval 69.1%

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

      3. *-commutative 69.1%

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

      4. unpow2 69.1%

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

      5. associate-*r* 69.2%

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

      6. *-commutative 69.2%

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

   6. Simplified 69.2%

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

   if -8.0000000000000003e71 < x.re < -7.0000000000000004e23 or -5.60000000000000008e-64 < x.re < 3.99999999999999989e59

   1. Initial program 93.9%

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

      1. expm1-log1p-u 67.2%

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

      2. expm1-udef 48.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 92.5%

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

   6. Taylor expanded in x.re around 0 91.5%

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

   8. Simplified 91.5%

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

   if -7.0000000000000004e23 < x.re < -5.60000000000000008e-64 or 3.99999999999999989e59 < x.re

   1. Initial program 78.5%

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

   2. Taylor expanded in x.im around 0 72.4%

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

   4. Simplified 72.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8 \cdot 10^{+71}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq -7 \cdot 10^{+23} \lor \neg \left(x.re \leq -5.6 \cdot 10^{-64}\right) \land x.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 6: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.65 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -5.6 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* 3.0 (* x.re x.im)))) (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= x.re -7.5e+71)
     t_0
     (if (<= x.re -2.65e+23)
       t_1
       (if (<= x.re -5.6e-64)
         (* x.im (* 3.0 (* x.re x.re)))
         (if (<= x.re 2.85e+60) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (x_46_re <= -7.5e+71) {
		tmp = t_0;
	} else if (x_46_re <= -2.65e+23) {
		tmp = t_1;
	} else if (x_46_re <= -5.6e-64) {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	} else if (x_46_re <= 2.85e+60) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 * (3.0d0 * (x_46re * x_46im))
    t_1 = x_46im * (x_46im * -x_46im)
    if (x_46re <= (-7.5d+71)) then
        tmp = t_0
    else if (x_46re <= (-2.65d+23)) then
        tmp = t_1
    else if (x_46re <= (-5.6d-64)) then
        tmp = x_46im * (3.0d0 * (x_46re * x_46re))
    else if (x_46re <= 2.85d+60) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (x_46_re <= -7.5e+71) {
		tmp = t_0;
	} else if (x_46_re <= -2.65e+23) {
		tmp = t_1;
	} else if (x_46_re <= -5.6e-64) {
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	} else if (x_46_re <= 2.85e+60) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * (3.0 * (x_46_re * x_46_im))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if x_46_re <= -7.5e+71:
		tmp = t_0
	elif x_46_re <= -2.65e+23:
		tmp = t_1
	elif x_46_re <= -5.6e-64:
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re))
	elif x_46_re <= 2.85e+60:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (x_46_re <= -7.5e+71)
		tmp = t_0;
	elseif (x_46_re <= -2.65e+23)
		tmp = t_1;
	elseif (x_46_re <= -5.6e-64)
		tmp = Float64(x_46_im * Float64(3.0 * Float64(x_46_re * x_46_re)));
	elseif (x_46_re <= 2.85e+60)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (x_46_re <= -7.5e+71)
		tmp = t_0;
	elseif (x_46_re <= -2.65e+23)
		tmp = t_1;
	elseif (x_46_re <= -5.6e-64)
		tmp = x_46_im * (3.0 * (x_46_re * x_46_re));
	elseif (x_46_re <= 2.85e+60)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -7.5e+71], t$95$0, If[LessEqual[x$46$re, -2.65e+23], t$95$1, If[LessEqual[x$46$re, -5.6e-64], N[(x$46$im * N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.85e+60], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.50000000000000007e71 or 2.84999999999999989e60 < x.re

   1. Initial program 60.5%

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

      1. add-log-exp 38.6%

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

      2. *-un-lft-identity 38.6%

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

      3. log-prod 38.6%

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

      4. metadata-eval 38.6%

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

      5. add-log-exp 60.5%

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

      6. *-commutative 60.5%

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

      7. difference-of-squares 70.5%

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

      8. associate-*r* 81.9%

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

   3. Applied egg-rr 81.9%

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

   4. Taylor expanded in x.im around 0 70.5%

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

      1. distribute-lft1-in 70.5%

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

      2. metadata-eval 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-*r* 70.5%

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

      5. metadata-eval 70.5%

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

      6. distribute-lft1-in 70.5%

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

      7. unpow2 70.5%

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

      8. associate-*l* 81.8%

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

      9. distribute-lft1-in 81.8%

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

      10. metadata-eval 81.8%

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

      11. *-commutative 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in x.re around 0 81.8%

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

   if -7.50000000000000007e71 < x.re < -2.6500000000000001e23 or -5.60000000000000008e-64 < x.re < 2.84999999999999989e60

   1. Initial program 93.9%

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

      1. expm1-log1p-u 67.2%

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

      2. expm1-udef 48.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 92.5%

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

   6. Taylor expanded in x.re around 0 91.5%

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

   8. Simplified 91.5%

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

   if -2.6500000000000001e23 < x.re < -5.60000000000000008e-64

   1. Initial program 99.7%

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

   2. Taylor expanded in x.im around 0 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+71}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -2.65 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -5.6 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 7: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;x.re \leq -8.2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.7 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -5.6 \cdot 10^{-64}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* 3.0 (* x.re x.im)))) (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= x.re -8.2e+71)
     t_0
     (if (<= x.re -2.7e+23)
       t_1
       (if (<= x.re -5.6e-64)
         (* (* x.re x.re) (* x.im 3.0))
         (if (<= x.re 4e+59) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (x_46_re <= -8.2e+71) {
		tmp = t_0;
	} else if (x_46_re <= -2.7e+23) {
		tmp = t_1;
	} else if (x_46_re <= -5.6e-64) {
		tmp = (x_46_re * x_46_re) * (x_46_im * 3.0);
	} else if (x_46_re <= 4e+59) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 * (3.0d0 * (x_46re * x_46im))
    t_1 = x_46im * (x_46im * -x_46im)
    if (x_46re <= (-8.2d+71)) then
        tmp = t_0
    else if (x_46re <= (-2.7d+23)) then
        tmp = t_1
    else if (x_46re <= (-5.6d-64)) then
        tmp = (x_46re * x_46re) * (x_46im * 3.0d0)
    else if (x_46re <= 4d+59) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (x_46_re <= -8.2e+71) {
		tmp = t_0;
	} else if (x_46_re <= -2.7e+23) {
		tmp = t_1;
	} else if (x_46_re <= -5.6e-64) {
		tmp = (x_46_re * x_46_re) * (x_46_im * 3.0);
	} else if (x_46_re <= 4e+59) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * (3.0 * (x_46_re * x_46_im))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if x_46_re <= -8.2e+71:
		tmp = t_0
	elif x_46_re <= -2.7e+23:
		tmp = t_1
	elif x_46_re <= -5.6e-64:
		tmp = (x_46_re * x_46_re) * (x_46_im * 3.0)
	elif x_46_re <= 4e+59:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (x_46_re <= -8.2e+71)
		tmp = t_0;
	elseif (x_46_re <= -2.7e+23)
		tmp = t_1;
	elseif (x_46_re <= -5.6e-64)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im * 3.0));
	elseif (x_46_re <= 4e+59)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (3.0 * (x_46_re * x_46_im));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (x_46_re <= -8.2e+71)
		tmp = t_0;
	elseif (x_46_re <= -2.7e+23)
		tmp = t_1;
	elseif (x_46_re <= -5.6e-64)
		tmp = (x_46_re * x_46_re) * (x_46_im * 3.0);
	elseif (x_46_re <= 4e+59)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -8.2e+71], t$95$0, If[LessEqual[x$46$re, -2.7e+23], t$95$1, If[LessEqual[x$46$re, -5.6e-64], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4e+59], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8.2000000000000004e71 or 3.99999999999999989e59 < x.re

   1. Initial program 60.5%

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

      1. add-log-exp 38.6%

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

      2. *-un-lft-identity 38.6%

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

      3. log-prod 38.6%

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

      4. metadata-eval 38.6%

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

      5. add-log-exp 60.5%

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

      6. *-commutative 60.5%

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

      7. difference-of-squares 70.5%

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

      8. associate-*r* 81.9%

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

   3. Applied egg-rr 81.9%

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

   4. Taylor expanded in x.im around 0 70.5%

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

      1. distribute-lft1-in 70.5%

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

      2. metadata-eval 70.5%

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

      3. *-commutative 70.5%

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

      4. associate-*r* 70.5%

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

      5. metadata-eval 70.5%

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

      6. distribute-lft1-in 70.5%

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

      7. unpow2 70.5%

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

      8. associate-*l* 81.8%

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

      9. distribute-lft1-in 81.8%

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

      10. metadata-eval 81.8%

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

      11. *-commutative 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in x.re around 0 81.8%

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

   if -8.2000000000000004e71 < x.re < -2.6999999999999999e23 or -5.60000000000000008e-64 < x.re < 3.99999999999999989e59

   1. Initial program 93.9%

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

      1. expm1-log1p-u 67.2%

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

      2. expm1-udef 48.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 92.5%

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

   6. Taylor expanded in x.re around 0 91.5%

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

   8. Simplified 91.5%

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

   if -2.6999999999999999e23 < x.re < -5.60000000000000008e-64

   1. Initial program 99.7%

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

   2. Taylor expanded in x.re around inf 72.7%

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

   4. Simplified 72.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8.2 \cdot 10^{+71}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -2.7 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.re \leq -5.6 \cdot 10^{-64}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{-48} \lor \neg \left(x.im \leq 9.5 \cdot 10^{-97}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -7.8e-48) (not (<= x.im 9.5e-97)))
   (* x.im (- (* x.re x.re) (* x.im x.im)))
   (* x.re (* 3.0 (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -7.8e-48) || !(x_46_im <= 9.5e-97)) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

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 <= (-7.8d-48)) .or. (.not. (x_46im <= 9.5d-97))) then
        tmp = x_46im * ((x_46re * x_46re) - (x_46im * x_46im))
    else
        tmp = x_46re * (3.0d0 * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -7.8e-48) || !(x_46_im <= 9.5e-97)) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -7.8e-48) or not (x_46_im <= 9.5e-97):
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -7.8e-48) || !(x_46_im <= 9.5e-97))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -7.8e-48) || ~((x_46_im <= 9.5e-97)))
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	else
		tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -7.8e-48], N[Not[LessEqual[x$46$im, 9.5e-97]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -7.800000000000001e-48 or 9.5000000000000001e-97 < x.im

   1. Initial program 77.4%

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

      1. expm1-log1p-u 43.5%

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

      2. expm1-udef 34.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 81.5%

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

   if -7.800000000000001e-48 < x.im < 9.5000000000000001e-97

   1. Initial program 88.0%

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

      1. add-log-exp 50.1%

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

      2. *-un-lft-identity 50.1%

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

      3. log-prod 50.1%

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

      4. metadata-eval 50.1%

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

      5. add-log-exp 88.0%

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

      6. *-commutative 88.0%

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

      7. difference-of-squares 88.0%

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

      8. associate-*r* 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x.im around 0 82.1%

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

      1. distribute-lft1-in 82.1%

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

      2. metadata-eval 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r* 82.1%

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

      5. metadata-eval 82.1%

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

      6. distribute-lft1-in 82.1%

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

      7. unpow2 82.1%

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

      8. associate-*l* 93.9%

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

      9. distribute-lft1-in 93.9%

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

      10. metadata-eval 93.9%

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

      11. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in x.re around 0 93.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{-48} \lor \neg \left(x.im \leq 9.5 \cdot 10^{-97}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 9: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{+153} \lor \neg \left(x.re \leq 4.5 \cdot 10^{+149}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -7e+153) (not (<= x.re 4.5e+149)))
   (* x.re (* x.re x.im))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7e+153) || !(x_46_re <= 4.5e+149)) {
		tmp = x_46_re * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

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 <= (-7d+153)) .or. (.not. (x_46re <= 4.5d+149))) then
        tmp = x_46re * (x_46re * x_46im)
    else
        tmp = x_46im * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7e+153) || !(x_46_re <= 4.5e+149)) {
		tmp = x_46_re * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -7e+153) or not (x_46_re <= 4.5e+149):
		tmp = x_46_re * (x_46_re * x_46_im)
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -7e+153) || !(x_46_re <= 4.5e+149))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -7e+153) || ~((x_46_re <= 4.5e+149)))
		tmp = x_46_re * (x_46_re * x_46_im);
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -7e+153], N[Not[LessEqual[x$46$re, 4.5e+149]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -6.9999999999999998e153 or 4.49999999999999982e149 < x.re

   1. Initial program 55.8%

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

      1. expm1-log1p-u 27.9%

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

      2. expm1-udef 27.9%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 54.5%

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

   6. Taylor expanded in x.re around inf 70.6%

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

   8. Simplified 70.6%

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

   9. Taylor expanded in x.im around 0 70.6%

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

   11. Simplified 73.1%

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

   if -6.9999999999999998e153 < x.re < 4.49999999999999982e149

   1. Initial program 89.5%

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

      1. expm1-log1p-u 65.0%

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

      2. expm1-udef 42.3%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

   5. Simplified 77.0%

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

   6. Taylor expanded in x.re around 0 72.8%

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

   8. Simplified 72.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{+153} \lor \neg \left(x.re \leq 4.5 \cdot 10^{+149}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 10: 33.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. expm1-log1p-u 56.0%

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

   2. expm1-udef 38.8%

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

3. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

5. Simplified 71.6%

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

6. Taylor expanded in x.re around inf 34.1%

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

8. Simplified 34.1%

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

9. Final simplification 34.1%

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

Alternative 11: 34.4% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.re (* 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);
}

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)
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);
}

Python

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

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(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);
end

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. expm1-log1p-u 56.0%

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

   2. expm1-udef 38.8%

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

3. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x.re, \frac{0}{0}, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\right)} - 1} \]

5. Simplified 71.6%

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

6. Taylor expanded in x.re around inf 34.1%

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

8. Simplified 34.1%

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

9. Taylor expanded in x.im around 0 34.1%

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

11. Simplified 34.6%

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

12. Final simplification 34.6%

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

Developer target: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023174 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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