Average Error: 25.8 → 17.9

Time: 14.7s

Precision: binary64

Cost: 27272

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.6836303645104744 \cdot 10^{+137} \lor \neg \left(y.im \leq 4.0463340212107907 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}

↓

\begin{array}{l}
\mathbf{if}\;y.im \leq -3.6836303645104744 \cdot 10^{+137} \lor \neg \left(y.im \leq 4.0463340212107907 \cdot 10^{+55}\right):\\
\;\;\;\;\frac{x.im}{y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}

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

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.6836303645104744e+137)
         (not (<= y.im 4.0463340212107907e+55)))
   (/ x.im y.im)
   (/
    (/
     (+ (* y.re x.re) (* y.im x.im))
     (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
    (sqrt (+ (* y.re y.re) (* y.im y.im))))))

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * 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 tmp;
	if ((y_46_im <= -3.6836303645104744e+137) || !(y_46_im <= 4.0463340212107907e+55)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (((y_46_re * x_46_re) + (y_46_im * x_46_im)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x.re`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	26.4
Cost	79424

\[\frac{\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im} \cdot \sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 2
Error	45.0
Cost	72512

\[\frac{\sqrt{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 3
Error	25.9
Cost	72128

\[\frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \sqrt{\sqrt{{y.re}^{2} + {y.im}^{2}}}}\]

Alternative 4
Error	26.4
Cost	60352

\[\frac{1}{\sqrt[3]{\frac{{y.re}^{2} + {y.im}^{2}}{y.re \cdot x.re + y.im \cdot x.im}} \cdot \left(\sqrt[3]{\frac{{y.re}^{2} + {y.im}^{2}}{y.re \cdot x.re + y.im \cdot x.im}} \cdot \sqrt[3]{\frac{{y.re}^{2} + {y.im}^{2}}{y.re \cdot x.re + y.im \cdot x.im}}\right)}\]

Alternative 5
Error	26.2
Cost	60224

\[\sqrt[3]{\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}} \cdot \left(\sqrt[3]{\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}} \cdot \sqrt[3]{\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}}\right)\]

Alternative 6
Error	26.2
Cost	59840

\[\frac{\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im} \cdot \sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 7
Error	26.3
Cost	59328

\[\frac{1}{\left(\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}\right) \cdot \frac{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}{y.re \cdot x.re + y.im \cdot x.im}}\]

Alternative 8
Error	26.3
Cost	59328

\[\frac{1}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 9
Error	44.8
Cost	52928

\[\frac{\sqrt{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt{y.re \cdot x.re + y.im \cdot x.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 10
Error	51.7
Cost	52672

\[\frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \sqrt{y.re}}\]

Alternative 11
Error	26.3
Cost	46528

\[\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}}}{\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}}\]

Alternative 12
Error	40.0
Cost	40128

\[\sqrt{\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}} \cdot \sqrt{\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 13
Error	25.8
Cost	39744

\[\frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 14
Error	26.3
Cost	33856

\[\left(\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im} \cdot \sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}\right) \cdot \frac{\sqrt[3]{y.re \cdot x.re + y.im \cdot x.im}}{{y.re}^{2} + {y.im}^{2}}\]

Alternative 15
Error	44.8
Cost	26944

\[\sqrt{y.re \cdot x.re + y.im \cdot x.im} \cdot \frac{\sqrt{y.re \cdot x.re + y.im \cdot x.im}}{{y.re}^{2} + {y.im}^{2}}\]

Alternative 16
Error	25.7
Cost	26944

\[\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

Alternative 17
Error	40.9
Cost	26496

\[\sqrt[3]{{\left(\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}\right)}^{3}}\]

Alternative 18
Error	41.0
Cost	26432

\[e^{\log \left(\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}\right)}\]

Alternative 19
Error	44.2
Cost	20992

\[\frac{1}{\left(\frac{y.re}{x.re} + \frac{{y.im}^{2}}{y.re} \cdot \frac{{x.im}^{2}}{{x.re}^{3}}\right) + \frac{y.im}{x.re} \cdot \left(\frac{y.im}{y.re} - \frac{x.im}{x.re}\right)}\]

Alternative 20
Error	38.9
Cost	20992

\[\frac{1}{\frac{y.re}{y.im} \cdot \left(\frac{y.re}{x.im} + \frac{y.re \cdot {x.re}^{2}}{{x.im}^{3}}\right) + \left(\frac{y.im}{x.im} - \frac{y.re \cdot x.re}{{x.im}^{2}}\right)}\]

Alternative 21
Error	47.4
Cost	20352

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

Alternative 22
Error	46.7
Cost	20352

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

Alternative 23
Error	54.7
Cost	15168

\[\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{6} + {y.im}^{6}} \cdot \left(\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)\right)\]

Alternative 24
Error	51.8
Cost	14656

\[\frac{\frac{{y.re}^{2} \cdot \left(x.re \cdot x.re\right) - {y.im}^{2} \cdot \left(x.im \cdot x.im\right)}{y.re \cdot x.re - y.im \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im}\]

Alternative 25
Error	41.5
Cost	14656

\[\frac{\left(y.re \cdot x.re\right) \cdot \left(y.re \cdot x.re\right) - \left(y.im \cdot x.im\right) \cdot \left(y.im \cdot x.im\right)}{\left({y.re}^{2} + {y.im}^{2}\right) \cdot \left(y.re \cdot x.re - y.im \cdot x.im\right)}\]

Alternative 26
Error	44.8
Cost	14272

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

Alternative 27
Error	48.7
Cost	14144

\[\frac{y.re \cdot x.re + y.im \cdot x.im}{{y.re}^{4} - {y.im}^{4}} \cdot \left(y.re \cdot y.re - y.im \cdot y.im\right)\]

Alternative 28
Error	45.5
Cost	13824

\[\frac{\sqrt[3]{{\left(y.re \cdot x.re + y.im \cdot x.im\right)}^{3}}}{y.re \cdot y.re + y.im \cdot y.im}\]

Alternative 29
Error	26.0
Cost	13760

\[\left(y.re \cdot x.re + y.im \cdot x.im\right) \cdot \frac{1}{{y.re}^{2} + {y.im}^{2}}\]

Alternative 30
Error	25.9
Cost	13760

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

Alternative 31
Error	39.1
Cost	13504

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

Alternative 32
Error	39.5
Cost	13504

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

Alternative 33
Error	39.0
Cost	13376

\[\frac{y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}\]

Alternative 34
Error	39.4
Cost	13376

\[\frac{y.re \cdot x.re}{{y.re}^{2} + {y.im}^{2}}\]

Alternative 35
Error	25.8
Cost	960

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

Alternative 36
Error	39.4
Cost	704

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

Alternative 37
Error	39.0
Cost	704

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

Alternative 38
Error	37.9
Cost	320

\[\frac{1}{\frac{y.re}{x.re}}\]

Alternative 39
Error	37.1
Cost	320

\[\frac{1}{\frac{y.im}{x.im}}\]

Alternative 40
Error	37.8
Cost	192

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

Alternative 41
Error	36.9
Cost	192

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

Alternative 42
Error	61.6
Cost	64

\[1\]

Alternative 43
Error	52.3
Cost	64

\[0\]

Alternative 44
Error	61.6
Cost	64

\[-1\]

Error

Derivation

Split input into 2 regimes
if y.im < -3.6836303645104744e137 or 4.0463340212107907e55 < y.im
1. Initial program 38.6
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Taylor expanded around 0 17.3
  \[\leadsto \color{blue}{\frac{x.im}{y.im}}\]
3. Simplified17.3
  \[\leadsto \color{blue}{\frac{x.im}{y.im}}\]
if -3.6836303645104744e137 < y.im < 4.0463340212107907e55
1. Initial program 18.4
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt_binary64_78218.4
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied associate-/r*_binary64_70418.3
  \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
5. Simplified18.3
  \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
6. Simplified18.3
  \[\leadsto \color{blue}{\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
Recombined 2 regimes into one program.
Final simplification17.9
\[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6836303645104744 \cdot 10^{+137} \lor \neg \left(y.im \leq 4.0463340212107907 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

Error

Try it out

Alternatives

Error

Derivation

`if y.im < -3.6836303645104744e137 or 4.0463340212107907e55 < y.im`

`if -3.6836303645104744e137 < y.im < 4.0463340212107907e55`

Reproduce