math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 96.7%

Time: 6.1s

Precision: binary64

Cost: 7492

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

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} x.re = |x.re|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

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)))

↓

NOTE: x.re should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.6e+153)
   (* x.im (- (* 3.0 (* x.re x.re)) (* x.im x.im)))
   (fma (* x.im (* x.re 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_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);
}

↓

x.re = abs(x.re);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.6e+153) {
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	} else {
		tmp = fma((x_46_im * (x_46_re * 2.0)), x_46_re, ((x_46_im * (x_46_re + x_46_im)) * (x_46_re - x_46_im)));
	}
	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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

↓

x.re = abs(x.re)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.6e+153)
		tmp = Float64(x_46_im * Float64(Float64(3.0 * Float64(x_46_re * x_46_re)) - Float64(x_46_im * x_46_im)));
	else
		tmp = fma(Float64(x_46_im * Float64(x_46_re * 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
	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$46$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]

↓

NOTE: x.re should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.6e+153], N[(x$46$im * N[(N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision] * x$46$re + N[(N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	82.7%
Target	91.6%
Herbie	96.7%

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.59999999999999933e153

   1. Initial program 87.7%

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

   2. Taylor expanded in x.re around 0 87.6%

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

   3. Simplified 87.6%

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

      [Start] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + 2 \cdot \left({x.re}^{2} \cdot x.im\right) \]
      associate-*r* [=>] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]
      *-commutative [=>] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]
      unpow2 [=>] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]
      associate-*r* [=>] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
      Step-by-step derivation

      [Start] 87.6%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) \]
      expm1-log1p-u [=>] 61.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]
      expm1-udef [=>] 42.0%	\[ \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]
      +-commutative [=>] 42.0%	\[ e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]
      *-commutative [=>] 42.0%	\[ e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]
      distribute-lft-out [=>] 46.1%	\[ e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]
      *-commutative [=>] 46.1%	\[ e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]
      *-commutative [=>] 46.1%	\[ e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   5. Simplified 94.1%

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

      [Start] 46.1%	\[ e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]
      expm1-def [=>] 65.6%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]
      expm1-log1p [=>] 94.0%	\[ \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
      unpow2 [<=] 94.0%	\[ x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]
      associate-+r- [=>] 94.1%	\[ x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]
      associate-*r* [=>] 94.1%	\[ x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]
      unpow2 [<=] 94.1%	\[ x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]
      *-commutative [<=] 94.1%	\[ x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]
      distribute-lft1-in [=>] 94.1%	\[ x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]
      metadata-eval [=>] 94.1%	\[ x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]
      unpow2 [=>] 94.1%	\[ x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   if 7.59999999999999933e153 < x.re

   1. Initial program 48.0%

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

   2. Taylor expanded in x.re around 0 48.0%

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

   3. Simplified 48.0%

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

      [Start] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + 2 \cdot \left({x.re}^{2} \cdot x.im\right) \]
      associate-*r* [=>] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]
      *-commutative [=>] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]
      unpow2 [=>] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]
      associate-*r* [=>] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Applied egg-rr 89.1%

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

      [Start] 48.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) \]
      +-commutative [=>] 48.0%	\[ \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
      associate-*r* [=>] 48.0%	\[ \color{blue}{\left(x.im \cdot \left(2 \cdot x.re\right)\right) \cdot x.re} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]
      fma-def [=>] 48.0%	\[ \color{blue}{\mathsf{fma}\left(x.im \cdot \left(2 \cdot x.re\right), x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 48.0%	\[ \mathsf{fma}\left(x.im \cdot \color{blue}{\left(x.re \cdot 2\right)}, x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right) \]
      *-commutative [=>] 48.0%	\[ \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]
      difference-of-squares [=>] 64.2%	\[ \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, x.im \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]
      associate-*r* [=>] 89.1%	\[ \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.7%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x.re \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	96.7%
Cost	836

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

Alternative 3
Accuracy	77.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x.re \leq 6.1 \cdot 10^{-15}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	83.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x.re \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	83.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 6
Accuracy	59.9%
Cost	384

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

Alternative 7
Accuracy	2.7%
Cost	64

\[-3 \]

Reproduce

herbie shell --seed 2023171 
(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)))