_divideComplex, imaginary part

Percentage Accurate: 70.1% → 88.2%

Time: 29.9s

Precision: binary64

Cost: 14788

• could not determine a ground truth

  1. x.re = 2.387501152545511e-262
  2. x.im = 1.2178575715387347e-86
  3. y.re = -1.3729119915258474e+173
  4. y.im = -5.053792087175491e+68

Math

\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

↓

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Inf)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 82.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      Step-by-step derivation

      [Start] 82.6%	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 82.6%	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 82.6%	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 82.5%	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 82.6%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 94.9%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if +inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 73.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
      Step-by-step derivation

      [Start] 73.4%	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 73.4%	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 73.4%	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 73.4%	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      unpow2 [=>] 73.4%	\[ \frac{y.re \cdot x.im}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 73.8%	\[ \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im}} - \frac{x.re}{y.im} \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.2%
Cost	14788

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2
Accuracy	87.5%
Cost	13768

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-208}:\\ \;\;\;\;\frac{t_0}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	87.6%
Cost	7760

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-208}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	87.6%
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -5.5 \cdot 10^{-208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	80.5%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3000000000000 \lor \neg \left(y.im \leq -5.3 \cdot 10^{-40}\right) \land \left(y.im \leq -1.15 \cdot 10^{-69} \lor \neg \left(y.im \leq 4.6 \cdot 10^{-11}\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \end{array} \]

Alternative 6
Accuracy	82.1%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -12500000000000:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{x.re \cdot y.im}{y.re} - x.im\right)\\ \end{array} \]

Alternative 7
Accuracy	82.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -12500000000000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-86}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{x.re \cdot y.im}{y.re} - x.im\right)\\ \end{array} \]

Alternative 8
Accuracy	82.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -13000000000000:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im \cdot \frac{x.re}{y.re}}}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{x.re \cdot y.im}{y.re} - x.im\right)\\ \end{array} \]

Alternative 9
Accuracy	69.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-71} \lor \neg \left(y.im \leq 2.75 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	43.7%
Cost	192

\[\frac{x.im}{y.re} \]

Reproduce

herbie shell --seed 2023277 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))