math.cube on complex, imaginary part

Average Accuracy: 49.4% → 55.8%

Time: 9.0s

Precision: binary64

Cost: 3785

Math

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

↓

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := t_0 + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;t_0 + \left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_0 (* x.re (+ (* x.re x.im) (* x.re x.im))))
        (t_1 (+ t_0 (* x.im (- (* x.re x.re) (* x.im x.im))))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+303)))
     (+ t_0 (* (* x.re x.im) (- x.re x.im)))
     t_1)))

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 t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+303)) {
		tmp = t_0 + ((x_46_re * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = t_1;
	}
	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 t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+303)) {
		tmp = t_0 + ((x_46_re * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = t_1;
	}
	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):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+303):
		tmp = t_0 + ((x_46_re * x_46_im) * (x_46_re - x_46_im))
	else:
		tmp = t_1
	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)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	t_1 = Float64(t_0 + Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+303))
		tmp = Float64(t_0 + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - x_46_im)));
	else
		tmp = t_1;
	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)
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+303)))
		tmp = t_0 + ((x_46_re * x_46_im) * (x_46_re - x_46_im));
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+303]], $MachinePrecision]], N[(t$95$0 + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	49.4%
Target	55.5%
Herbie	55.8%

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

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 or 4.9999999999999997e303 < (+.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 13.8%

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

      [Start] 0.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      add-log-exp [=>] 0.0	\[ \color{blue}{\log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-un-lft-identity [=>] 0.0	\[ \log \color{blue}{\left(1 \cdot e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      log-prod [=>] 0.0	\[ \color{blue}{\left(\log 1 + \log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      metadata-eval [=>] 0.0	\[ \left(\color{blue}{0} + \log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      add-log-exp [<=] 0.0	\[ \left(0 + \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 0.0	\[ \left(0 + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      difference-of-squares [=>] 0.0	\[ \left(0 + x.im \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      associate-*r* [=>] 13.8	\[ \left(0 + \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   3. Taylor expanded in x.im around 0 14.3%

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

   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)) < 4.9999999999999997e303

   1. Initial program 99.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) \leq -\infty \lor \neg \left(x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	55.5%
Cost	7040

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

Alternative 2
Accuracy	55.5%
Cost	7040

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

Alternative 3
Accuracy	55.6%
Cost	3785

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	55.5%
Cost	1216

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

Alternative 5
Accuracy	44.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{-76} \lor \neg \left(x.re \leq 6 \cdot 10^{-69}\right):\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	50.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.re \leq -3.55 \cdot 10^{-81} \lor \neg \left(x.re \leq 1.8 \cdot 10^{-68}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	50.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.4 \cdot 10^{-75}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.35 \cdot 10^{-68}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	50.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -9 \cdot 10^{-76}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 1.65 \cdot 10^{-68}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	32.8%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.3 \cdot 10^{-95} \lor \neg \left(x.im \leq 1.1 \cdot 10^{-88}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 10
Accuracy	17.1%
Cost	320

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

Alternative 11
Accuracy	18.1%
Cost	320

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

Alternative 12
Accuracy	4.5%
Cost	128

\[-x.im \]

Alternative 13
Accuracy	2.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023157 
(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)))