math.cube on complex, real part

Average Accuracy: 49.5% → 54.2%

Time: 10.0s

Precision: binary64

Cost: 3657

Math

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

↓

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ t_1 := t_0 - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_2 := x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 4 \cdot 10^{+228}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_2\\ \end{array} \]

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
 (let* ((t_0 (* x.re (- (* x.re x.re) (* x.im x.im))))
        (t_1 (- t_0 (* x.im (+ (* x.re x.im) (* x.re x.im)))))
        (t_2 (* x.im (* x.re (+ x.im x.im)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 4e+228)))
     (- (* (* x.re x.im) (- x.re x.im)) t_2)
     (- t_0 t_2))))

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 t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_2 = x_46_im * (x_46_re * (x_46_im + x_46_im));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 4e+228)) {
		tmp = ((x_46_re * x_46_im) * (x_46_re - x_46_im)) - t_2;
	} else {
		tmp = t_0 - t_2;
	}
	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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_2 = x_46_im * (x_46_re * (x_46_im + x_46_im));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 4e+228)) {
		tmp = ((x_46_re * x_46_im) * (x_46_re - x_46_im)) - t_2;
	} else {
		tmp = t_0 - t_2;
	}
	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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

↓

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	t_1 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_2 = x_46_im * (x_46_re * (x_46_im + x_46_im))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 4e+228):
		tmp = ((x_46_re * x_46_im) * (x_46_re - x_46_im)) - t_2
	else:
		tmp = t_0 - t_2
	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)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)))
	t_1 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_2 = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 4e+228))
		tmp = Float64(Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - x_46_im)) - t_2);
	else
		tmp = Float64(t_0 - t_2);
	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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

↓

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	t_1 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_2 = x_46_im * (x_46_re * (x_46_im + x_46_im));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 4e+228)))
		tmp = ((x_46_re * x_46_im) * (x_46_re - x_46_im)) - t_2;
	else
		tmp = t_0 - t_2;
	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$46$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_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 4e+228]], $MachinePrecision]], N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$0 - t$95$2), $MachinePrecision]]]]]

Target

Original	49.5%
Target	55.3%
Herbie	54.2%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -inf.0 or 3.9999999999999997e228 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 1.5%

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

   2. Simplified 1.5%

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

      [Start] 1.5	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 1.5	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 1.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      *-commutative [<=] 1.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      distribute-lft-out [=>] 1.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 12.1%

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

      [Start] 1.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      add-log-exp [=>] 0.0	\[ \color{blue}{\log \left(e^{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      *-un-lft-identity [=>] 0.0	\[ \log \color{blue}{\left(1 \cdot e^{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      log-prod [=>] 0.0	\[ \color{blue}{\left(\log 1 + \log \left(e^{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      metadata-eval [=>] 0.0	\[ \left(\color{blue}{0} + \log \left(e^{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      add-log-exp [<=] 1.5	\[ \left(0 + \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      difference-of-squares [=>] 1.5	\[ \left(0 + x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      associate-*r* [=>] 12.1	\[ \left(0 + \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   4. Taylor expanded in x.re around 0 12.8%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 3.9999999999999997e228

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 99.8	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 99.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      *-commutative [<=] 99.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      distribute-lft-out [=>] 99.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -\infty \lor \neg \left(x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 4 \cdot 10^{+228}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	55.3%
Cost	13312

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

Alternative 2
Accuracy	55.1%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+78}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 3
Accuracy	55.3%
Cost	1088

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

Alternative 4
Accuracy	50.6%
Cost	1028

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.75 \cdot 10^{-67}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	50.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.45 \cdot 10^{-65}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-21}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	44.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.7 \cdot 10^{-65} \lor \neg \left(x.im \leq 2 \cdot 10^{-22}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 7
Accuracy	50.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -7.5 \cdot 10^{-66} \lor \neg \left(x.im \leq 1.55 \cdot 10^{-21}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 8
Accuracy	15.4%
Cost	320

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

Alternative 9
Accuracy	30.6%
Cost	320

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

Reproduce

herbie shell --seed 2023157 
(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)))