Average Error: 25.7 → 14.2

Time: 10.0s

Precision: binary64

Cost: 28228

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.1437132202364462 \cdot 10^{+93}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5652367524611406 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.5711842677325443 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 1.400268501927579 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y.im \leq -1.1437132202364462 \cdot 10^{+93}:\\
\;\;\;\;\frac{-x.re}{y.im}\\

\mathbf{elif}\;y.im \leq -2.5652367524611406 \cdot 10^{-141}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 1.5711842677325443 \cdot 10^{-128}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 1.400268501927579 \cdot 10^{+125}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x.re}{y.im}\\

\end{array}

(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
 (if (<= y.im -1.1437132202364462e+93)
   (/ (- x.re) y.im)
   (if (<= y.im -2.5652367524611406e-141)
     (/
      (/
       (- (* y.re x.im) (* y.im x.re))
       (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 1.5711842677325443e-128)
       (- (/ x.im y.re) (/ (* y.im x.re) (pow y.re 2.0)))
       (if (<= y.im 1.400268501927579e+125)
         (/
          (/
           (- (* y.re x.im) (* y.im x.re))
           (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
          (sqrt (+ (* y.re y.re) (* y.im y.im))))
         (/ (- x.re) y.im))))))

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 tmp;
	if (y_46_im <= -1.1437132202364462e+93) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_im <= -2.5652367524611406e-141) {
		tmp = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / 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));
	} else if (y_46_im <= 1.5711842677325443e-128) {
		tmp = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 1.400268501927579e+125) {
		tmp = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / 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));
	} else {
		tmp = -x_46_re / 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	26.3
Cost	79424

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

Alternative 2
Error	45.0
Cost	72512

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

Alternative 3
Error	26.2
Cost	60224

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

Alternative 4
Error	26.2
Cost	59840

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

Alternative 5
Error	26.2
Cost	59328

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

Alternative 6
Error	44.8
Cost	52928

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

Alternative 7
Error	27.5
Cost	52416

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

Alternative 8
Error	39.6
Cost	40128

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

Alternative 9
Error	25.7
Cost	39744

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

Alternative 10
Error	26.2
Cost	33856

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

Alternative 11
Error	47.1
Cost	28096

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

Alternative 12
Error	25.7
Cost	26944

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

Alternative 13
Error	44.8
Cost	26944

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

Alternative 14
Error	40.9
Cost	26496

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

Alternative 15
Error	26.2
Cost	21184

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

Alternative 16
Error	46.2
Cost	20160

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

Alternative 17
Error	54.7
Cost	15168

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

Alternative 18
Error	40.9
Cost	14656

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

Alternative 19
Error	51.7
Cost	14656

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

Alternative 20
Error	44.8
Cost	14272

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

Alternative 21
Error	48.5
Cost	14144

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

Alternative 22
Error	44.9
Cost	13824

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

Alternative 23
Error	25.9
Cost	13760

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

Alternative 24
Error	25.9
Cost	13760

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

Alternative 25
Error	38.9
Cost	13440

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

Alternative 26
Error	38.8
Cost	13376

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

Alternative 27
Error	47.2
Cost	7552

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

Alternative 28
Error	47.1
Cost	7488

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

Alternative 29
Error	46.3
Cost	7360

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

Alternative 30
Error	46.1
Cost	7360

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

Alternative 31
Error	46.7
Cost	7040

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

Alternative 32
Error	47.1
Cost	7040

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

Alternative 33
Error	35.2
Cost	7040

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

Alternative 34
Error	25.7
Cost	960

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

Alternative 35
Error	38.9
Cost	768

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

Alternative 36
Error	38.8
Cost	704

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

Alternative 37
Error	37.0
Cost	256

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

Alternative 38
Error	37.7
Cost	192

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

Alternative 39
Error	61.6
Cost	64

\[1\]

Alternative 40
Error	51.6
Cost	64

\[0\]

Alternative 41
Error	61.7
Cost	64

\[-1\]

Derivation

Split input into 3 regimes
if y.im < -1.1437132202364462e93 or 1.40026850192757901e125 < y.im
1. Initial program 39.4
  \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Taylor expanded around 0 15.3
  \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}}\]
3. Simplified15.3
  \[\leadsto \color{blue}{\frac{-x.re}{y.im}}\]
4. Simplified15.3
  \[\leadsto \color{blue}{\frac{-x.re}{y.im}}\]
if -1.1437132202364462e93 < y.im < -2.56523675246114059e-141 or 1.57118426773254431e-128 < y.im < 1.40026850192757901e125
1. Initial program 15.8
  \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt_binary64_75915.8
  \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied associate-/r*_binary64_68115.8
  \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
5. Simplified15.8
  \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
6. Simplified15.8
  \[\leadsto \color{blue}{\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
if -2.56523675246114059e-141 < y.im < 1.57118426773254431e-128
1. Initial program 23.2
  \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Taylor expanded around inf 10.2
  \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}}\]
3. Simplified10.2
  \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}}\]
4. Simplified10.2
  \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}}\]
Recombined 3 regimes into one program.
Final simplification14.2
\[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1437132202364462 \cdot 10^{+93}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5652367524611406 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.5711842677325443 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 1.400268501927579 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))

Error

Try it out

Alternatives

Error

Derivation

`if y.im < -1.1437132202364462e93 or 1.40026850192757901e125 < y.im`

`if -1.1437132202364462e93 < y.im < -2.56523675246114059e-141 or 1.57118426773254431e-128 < y.im < 1.40026850192757901e125`

`if -2.56523675246114059e-141 < y.im < 1.57118426773254431e-128`

Reproduce