math.cube on complex, imaginary part

Percentage Accurate: 81.9% → 99.3%

Time: 8.1s

Precision: binary64

Cost: 7305

• 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

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{+117} \lor \neg \left(x.im \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \end{array} \]

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 (or (<= x.im -1.25e+117) (not (<= x.im 4e+94)))
   (+ (* x.im (* (- x.re x.im) (+ x.im x.re))) (+ x.im x.im))
   (- (* (* x.re (* x.im x.re)) 3.0) (pow x.im 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_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 <= -1.25e+117) || !(x_46_im <= 4e+94)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} else {
		tmp = ((x_46_re * (x_46_im * x_46_re)) * 3.0) - pow(x_46_im, 3.0);
	}
	return tmp;
}

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

↓

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 <= (-1.25d+117)) .or. (.not. (x_46im <= 4d+94))) then
        tmp = (x_46im * ((x_46re - x_46im) * (x_46im + x_46re))) + (x_46im + x_46im)
    else
        tmp = ((x_46re * (x_46im * x_46re)) * 3.0d0) - (x_46im ** 3.0d0)
    end if
    code = tmp
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);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.25e+117) || !(x_46_im <= 4e+94)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} else {
		tmp = ((x_46_re * (x_46_im * x_46_re)) * 3.0) - Math.pow(x_46_im, 3.0);
	}
	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 <= -1.25e+117) or not (x_46_im <= 4e+94):
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im)
	else:
		tmp = ((x_46_re * (x_46_im * x_46_re)) * 3.0) - math.pow(x_46_im, 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_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 ((x_46_im <= -1.25e+117) || !(x_46_im <= 4e+94))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))) + Float64(x_46_im + x_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_im * x_46_re)) * 3.0) - (x_46_im ^ 3.0));
	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 <= -1.25e+117) || ~((x_46_im <= 4e+94)))
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	else
		tmp = ((x_46_re * (x_46_im * x_46_re)) * 3.0) - (x_46_im ^ 3.0);
	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[Or[LessEqual[x$46$im, -1.25e+117], N[Not[LessEqual[x$46$im, 4e+94]], $MachinePrecision]], N[(N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	81.9%
Target	90.8%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if x.im < -1.24999999999999996e117 or 4.0000000000000001e94 < x.im

   1. Initial program 68.9%

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

   2. Applied egg-rr 86.5%

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

      [Start] 68.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [<=] 68.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.re \]
      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-+ [<=] 79.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.im + x.im\right)} \cdot x.re \]
      *-commutative [<=] 79.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.im + x.im\right)} \]
      distribute-lft-in [=>] 79.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)} \]
      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-+ [<=] 86.5%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(x.im + x.im\right)} \]

   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] 86.5%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.im + x.im\right) \]
      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%	\[ \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + \left(x.im + x.im\right) \]

   if -1.24999999999999996e117 < x.im < 4.0000000000000001e94

   1. Initial program 92.1%

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

   2. 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] 92.1%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      +-commutative [=>] 92.1%	\[ \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
      *-commutative [=>] 92.1%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} \]
      sub-neg [=>] 92.1%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} \]
      distribute-lft-in [=>] 92.1%	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(-x.im \cdot x.im\right)\right)} \]
      associate-+r+ [=>] 92.1%	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + x.im \cdot \left(-x.im \cdot x.im\right)} \]
      distribute-rgt-neg-out [=>] 92.1%	\[ \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + \color{blue}{\left(-x.im \cdot \left(x.im \cdot x.im\right)\right)} \]
      unsub-neg [=>] 92.1%	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{+117} \lor \neg \left(x.im \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{+117} \lor \neg \left(x.im \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{+117} \lor \neg \left(x.im \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{+117} \lor \neg \left(x.im \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re \cdot 3\right) - {x.im}^{3}\\ \end{array} \]

Alternative 4
Accuracy	98.0%
Cost	1744

\[\begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ t_1 := x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -1.25 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7.8 \cdot 10^{-109}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 3000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	90.7%
Cost	1228

\[\begin{array}{l} t_0 := x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{if}\;x.im \leq -22000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.1 \cdot 10^{-19}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 310000:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	91.1%
Cost	1228

\[\begin{array}{l} t_0 := x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{if}\;x.im \leq -22000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.35 \cdot 10^{-82}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 200000000:\\ \;\;\;\;x.im \cdot x.re + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	79.0%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5.6 \cdot 10^{+167} \lor \neg \left(x.re \leq -2.1 \cdot 10^{+140}\right) \land \left(x.re \leq -1.1 \cdot 10^{+93} \lor \neg \left(x.re \leq 2.05 \cdot 10^{-82}\right)\right):\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 8
Accuracy	79.0%
Cost	976

\[\begin{array}{l} t_0 := x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right)\\ t_1 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -5.6 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -6 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -1.1 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	71.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5.8 \cdot 10^{+167} \lor \neg \left(x.re \leq 2.4 \cdot 10^{+124}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 10
Accuracy	35.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x.im \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot x.re\\ \end{array} \]

Alternative 11
Accuracy	20.3%
Cost	192

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

Alternative 12
Accuracy	2.7%
Cost	64

\[-3 \]

Reproduce

herbie shell --seed 2023165 
(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)))