Average Error: 32.4 → 18.1
Time: 6.1s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log re\right)\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log re\right)\right)\\

\end{array}
(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.8014590457402938e+51)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* (- (log 1.0) (* 2.0 (log (/ -1.0 re)))) (sqrt (/ 1.0 (log 10.0)))))
   (if (<= re 8.860843829360954e+93)
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (* (log (+ (* re re) (* im im))) (/ 1.0 (sqrt (log 10.0)))))
     (*
      (/ 0.5 (sqrt (log 10.0)))
      (* (sqrt (/ 1.0 (log 10.0))) (+ (log 1.0) (* 2.0 (log re))))))))
double code(double re, double im) {
	return (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) / ((double) log(10.0)));
}
↓
double code(double re, double im) {
	double tmp;
	if ((re <= -1.8014590457402938e+51)) {
		tmp = ((double) ((0.5 / ((double) sqrt(((double) log(10.0))))) * ((double) (((double) (((double) log(1.0)) - ((double) (2.0 * ((double) log((-1.0 / re))))))) * ((double) sqrt((1.0 / ((double) log(10.0)))))))));
	} else {
		double tmp_1;
		if ((re <= 8.860843829360954e+93)) {
			tmp_1 = ((double) ((0.5 / ((double) sqrt(((double) log(10.0))))) * ((double) (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) * (1.0 / ((double) sqrt(((double) log(10.0)))))))));
		} else {
			tmp_1 = ((double) ((0.5 / ((double) sqrt(((double) log(10.0))))) * ((double) (((double) sqrt((1.0 / ((double) log(10.0))))) * ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) log(re))))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -1.8014590457402938e51

    1. Initial program Error: 46.0 bits

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrtError: 46.0 bits

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/2Error: 46.0 bits

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-powError: 46.0 bits

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-fracError: 46.0 bits

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

    7. Taylor expanded around -inf Error: 11.2 bits

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

    if -1.8014590457402938e51 < re < 8.86084382936095357e93

    1. Initial program Error: 22.8 bits

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrtError: 22.8 bits

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/2Error: 22.8 bits

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-powError: 22.8 bits

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-fracError: 22.8 bits

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-invError: 22.7 bits

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    if 8.86084382936095357e93 < re

    1. Initial program Error: 50.7 bits

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrtError: 50.7 bits

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/2Error: 50.7 bits

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{0.5}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-powError: 50.7 bits

      \[\leadsto \frac{\color{blue}{0.5 \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-fracError: 50.7 bits

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

    7. Taylor expanded around inf Error: 9.5 bits

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

    8. SimplifiedError: 9.5 bits

      \[\leadsto \frac{0.5}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\log 1 + 2 \cdot \log re\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 18.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8014590457402938 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 - 2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq 8.860843829360954 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log re\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))