math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 97.1%

Time: 10.8s

Precision: binary64

Cost: 8324

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

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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)))

↓

(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.re x.im)) 3.0) (pow x.im 3.0))
   (+ (* x.im (* (- x.re x.im) (+ x.re x.im))) (+ x.im 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);
}

↓

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_re * x_46_im)) * 3.0) - pow(x_46_im, 3.0);
	} else {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_im + x_46_im);
	}
	return tmp;
}

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);
}

↓

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_re * x_46_im)) * 3.0) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_im + x_46_im);
	}
	return tmp;
}

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)

↓

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_re * x_46_im)) * 3.0) - math.pow(x_46_im, 3.0)
	else:
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_im + 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

↓

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 * Float64(x_46_re * x_46_im)) * 3.0) - (x_46_im ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im))) + Float64(x_46_im + x_46_im));
	end
	return tmp
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

↓

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_re * x_46_im)) * 3.0) - (x_46_im ^ 3.0);
	else
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_im + x_46_im);
	end
	tmp_2 = 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]

↓

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 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]

Target

Original	82.4%
Target	91.5%
Herbie	97.1%

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

Derivation

1. Split input into 2 regimes

2. if (+.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 91.3%

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

   2. Simplified 99.8%

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

      [Start] 91.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      +-commutative [=>] 91.3%	\[ \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
      *-commutative [=>] 91.3%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} \]
      sub-neg [=>] 91.3%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} \]
      distribute-lft-in [=>] 91.3%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(-x.im \cdot x.im\right)\right)} \]
      associate-+r+ [=>] 91.3%	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + x.im \cdot \left(-x.im \cdot x.im\right)} \]
      distribute-rgt-neg-out [=>] 91.3%	\[ \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + \color{blue}{\left(-x.im \cdot \left(x.im \cdot x.im\right)\right)} \]
      unsub-neg [=>] 91.3%	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)} \]

   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. Applied egg-rr 60.0%

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

      [Start] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [<=] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.re \]
      flip-+ [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}} \cdot x.re \]
      +-inverses [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]
      +-inverses [<=] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]
      +-inverses [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \cdot x.re \]
      +-inverses [<=] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \cdot x.re \]
      flip-+ [<=] 40.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.im + x.im\right)} \cdot x.re \]
      *-commutative [<=] 40.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.im + x.im\right)} \]
      distribute-lft-in [=>] 40.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)} \]
      flip-+ [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}} \]
      +-inverses [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]
      +-inverses [<=] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \]
      +-inverses [=>] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]
      +-inverses [<=] 0.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \]
      flip-+ [<=] 60.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.im + x.im\right)} \]

   3. Applied egg-rr 100.0%

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

      [Start] 60.0%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.im + x.im\right) \]
      difference-of-squares [=>] 100.0%	\[ \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + \left(x.im + x.im\right) \]
      *-commutative [=>] 100.0%	\[ \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + \left(x.im + x.im\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	97.1%
Cost	8324

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

Alternative 2
Accuracy	97.0%
Cost	8324

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

Alternative 3
Accuracy	97.0%
Cost	2116

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

Alternative 4
Accuracy	77.7%
Cost	977

\[\begin{array}{l} t_0 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -5.2 \cdot 10^{-66} \lor \neg \left(x.im \leq 4.8 \cdot 10^{+64}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\\ \end{array} \]

Alternative 5
Accuracy	96.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.8 \cdot 10^{+153} \lor \neg \left(x.re \leq 1.7 \cdot 10^{+151}\right):\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \end{array} \]

Alternative 6
Accuracy	74.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.9 \cdot 10^{+161}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 6.8 \cdot 10^{+24}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 7
Accuracy	72.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.55 \cdot 10^{+159} \lor \neg \left(x.re \leq 8 \cdot 10^{+164}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]

Alternative 8
Accuracy	34.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x.im \leq 6.4 \cdot 10^{+89}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.im\\ \end{array} \]

Alternative 9
Accuracy	35.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x.im \leq 6.4 \cdot 10^{+89}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.im\\ \end{array} \]

Alternative 10
Accuracy	20.1%
Cost	192

\[x.re \cdot x.im \]

Alternative 11
Accuracy	2.7%
Cost	64

\[-3 \]

Alternative 12
Accuracy	2.7%
Cost	64

\[0.1 \]

Reproduce

herbie shell --seed 2023263 
(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)))