math.cube on complex, real part

Percentage Accurate: 82.6% → 96.9%

Time: 11.3s

Precision: binary64

Cost: 13577

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

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

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.72 \cdot 10^{+105} \lor \neg \left(x.re \leq 2 \cdot 10^{+80}\right):\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.72e+105) (not (<= x.re 2e+80)))
   (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))
   (fma (* x.re x.im) (* x.im -3.0) (pow x.re 3.0))))

C

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

↓

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.72e+105) || !(x_46_re <= 2e+80)) {
		tmp = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = fma((x_46_re * x_46_im), (x_46_im * -3.0), pow(x_46_re, 3.0));
	}
	return tmp;
}

Julia

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

↓

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.72e+105) || !(x_46_re <= 2e+80))
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = fma(Float64(x_46_re * x_46_im), Float64(x_46_im * -3.0), (x_46_re ^ 3.0));
	end
	return tmp
end

Wolfram

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

↓

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.72e+105], N[Not[LessEqual[x$46$re, 2e+80]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	82.6%
Target	87.4%
Herbie	96.9%

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.7200000000000001e105 or 2e80 < x.re

   1. Initial program 75.0%

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

   2. Simplified 97.7%

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

      [Start] 75.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [<=] 75.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]
      distribute-lft-out [=>] 75.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
      associate-*l* [=>] 75.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 75.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      distribute-rgt-out-- [=>] 89.8	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      associate--l- [=>] 89.8	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      associate--l- [<=] 89.8	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      sub-neg [=>] 89.8	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      associate--l+ [=>] 89.8	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      fma-udef [<=] 97.7	\[ x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, \left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)}\right) \]
      associate-*r* [<=] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 - 2\right)\right)}\right) \]
      metadata-eval [=>] 97.7	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \]

   if -1.7200000000000001e105 < x.re < 2e80

   1. Initial program 90.3%

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

   2. Simplified 90.4%

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

      [Start] 90.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 90.3	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      sub-neg [=>] 90.3	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      distribute-lft-in [=>] 90.3	\[ \color{blue}{\left(x.re \cdot \left(x.re \cdot x.re\right) + x.re \cdot \left(-x.im \cdot x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      associate--l+ [=>] 90.3	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re\right) + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      cube-unmult [=>] 90.4	\[ \color{blue}{{x.re}^{3}} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      *-commutative [<=] 90.4	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im\right) \]
      distribute-lft-out [=>] 90.4	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im\right) \]
      associate-*l* [=>] 90.4	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)}\right) \]
      distribute-lft-out-- [=>] 90.4	\[ {x.re}^{3} + \color{blue}{x.re \cdot \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 90.4	\[ {x.re}^{3} + x.re \cdot \left(\color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 90.4	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 90.4	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 90.4	\[ {x.re}^{3} + x.re \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)\right)} \]
      associate-*l* [<=] 90.4	\[ {x.re}^{3} + \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(-1 - 2\right)} \]
      metadata-eval [=>] 90.4	\[ {x.re}^{3} + \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \color{blue}{-3} \]

   3. Applied egg-rr 99.7%

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

      [Start] 90.4	\[ {x.re}^{3} + \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3 \]
      +-commutative [=>] 90.4	\[ \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3 + {x.re}^{3}} \]
      associate-*r* [=>] 99.7	\[ \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot x.im\right)} \cdot -3 + {x.re}^{3} \]
      associate-*l* [=>] 99.7	\[ \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)} + {x.re}^{3} \]
      fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.72 \cdot 10^{+105} \lor \neg \left(x.re \leq 2 \cdot 10^{+80}\right):\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.0%
Cost	13577

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+69} \lor \neg \left(x.re \leq 2 \cdot 10^{+80}\right):\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot \left(x.re \cdot x.im\right), x.im, {x.re}^{3}\right)\\ \end{array} \]

Alternative 2
Accuracy	96.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -5.8 \cdot 10^{+166}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	77.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.45 \cdot 10^{+62} \lor \neg \left(x.im \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4
Accuracy	82.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{+62} \lor \neg \left(x.im \leq 2.7 \cdot 10^{+59}\right):\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5
Accuracy	82.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+62}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 8 \cdot 10^{+60}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	82.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.35 \cdot 10^{+62}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	23.5%
Cost	320

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

Alternative 8
Accuracy	59.4%
Cost	320

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

Reproduce

herbie shell --seed 2023160 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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