math.log10 on complex, real part

Average Error: 31.4 → 18.2

Time: 6.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.0802856481292018 \cdot 10^{+127}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq -1.8632042262810938 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 2.424827901233389 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log 1 + 2 \cdot \log im}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 5.509693746587087 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log re\right)\right)\\ \end{array}\]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0802856481292018e+127)
   (*
    (/ 0.5 (sqrt (log 10.0)))
    (* (+ (log 1.0) (* (log (/ -1.0 re)) -2.0)) (sqrt (/ 1.0 (log 10.0)))))
   (if (<= re -1.8632042262810938e-201)
     (*
      (sqrt (/ 0.5 (sqrt (log 10.0))))
      (*
       (sqrt (/ 0.5 (sqrt (log 10.0))))
       (/ (log (+ (* re re) (* im im))) (sqrt (log 10.0)))))
     (if (<= re 2.424827901233389e-276)
       (*
        (sqrt (/ 0.5 (sqrt (log 10.0))))
        (*
         (sqrt (/ 0.5 (sqrt (log 10.0))))
         (/ (+ (log 1.0) (* 2.0 (log im))) (sqrt (log 10.0)))))
       (if (<= re 5.509693746587087e+135)
         (*
          (sqrt 0.5)
          (/ (sqrt 0.5) (/ (log 10.0) (log (+ (* re re) (* im im))))))
         (*
          (/ 0.5 (sqrt (log 10.0)))
          (* (sqrt (/ 1.0 (log 10.0))) (+ (log 1.0) (* 2.0 (log re))))))))))

C

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 <= -1.0802856481292018e+127)) {
		tmp = ((double) ((0.5 / ((double) sqrt(((double) log(10.0))))) * ((double) (((double) (((double) log(1.0)) + ((double) (((double) log((-1.0 / re))) * -2.0)))) * ((double) sqrt((1.0 / ((double) log(10.0)))))))));
	} else {
		double tmp_1;
		if ((re <= -1.8632042262810938e-201)) {
			tmp_1 = ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * (((double) log(((double) (((double) (re * re)) + ((double) (im * im)))))) / ((double) sqrt(((double) log(10.0)))))))));
		} else {
			double tmp_2;
			if ((re <= 2.424827901233389e-276)) {
				tmp_2 = ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * ((double) (((double) sqrt((0.5 / ((double) sqrt(((double) log(10.0))))))) * (((double) (((double) log(1.0)) + ((double) (2.0 * ((double) log(im)))))) / ((double) sqrt(((double) log(10.0)))))))));
			} else {
				double tmp_3;
				if ((re <= 5.509693746587087e+135)) {
					tmp_3 = ((double) (((double) sqrt(0.5)) * (((double) sqrt(0.5)) / (((double) log(10.0)) / ((double) log(((double) (((double) (re * re)) + ((double) (im * im))))))))));
				} else {
					tmp_3 = ((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_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.0802856481292018e127

   1. Initial program Error: 56.9 bits

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

   3. Applied add-sqr-sqrt Error: 56.9 bits

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

   4. Applied pow1/2 Error: 56.9 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-pow Error: 56.9 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-frac Error: 56.9 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: 7.6 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. Simplified Error: 7.6 bits

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

   if -1.0802856481292018e127 < re < -1.86320422628109375e-201

   1. Initial program Error: 18.9 bits

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

   3. Applied add-sqr-sqrt Error: 18.9 bits

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

   4. Applied pow1/2 Error: 18.9 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-pow Error: 18.9 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-frac Error: 18.9 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 add-sqr-sqrt Error: 18.9 bits

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

   9. Applied associate-*l* Error: 18.8 bits

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

   10. Simplified Error: 18.8 bits

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

   if -1.86320422628109375e-201 < re < 2.4248279012333889e-276

   1. Initial program Error: 29.5 bits

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

   3. Applied add-sqr-sqrt Error: 29.5 bits

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

   4. Applied pow1/2 Error: 29.5 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-pow Error: 29.5 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-frac Error: 29.5 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 add-sqr-sqrt Error: 29.5 bits

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

   9. Applied associate-*l* Error: 29.4 bits

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

   10. Simplified Error: 29.4 bits

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

   11. Taylor expanded around 0 Error: 34.5 bits

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

   if 2.4248279012333889e-276 < re < 5.50969374658708718e135

   1. Initial program Error: 20.3 bits

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

   3. Applied add-sqr-sqrt Error: 20.3 bits

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

   4. Applied pow1/2 Error: 20.3 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-pow Error: 20.3 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-frac Error: 20.2 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 *-un-lft-identity Error: 20.2 bits

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

   9. Applied add-sqr-sqrt Error: 20.3 bits

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

   10. Applied times-frac Error: 20.2 bits

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

   11. Applied associate-*l* Error: 20.2 bits

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

   12. Simplified Error: 20.2 bits

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

   if 5.50969374658708718e135 < re

   1. Initial program Error: 59.6 bits

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

   3. Applied add-sqr-sqrt Error: 59.6 bits

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

   4. Applied pow1/2 Error: 59.6 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-pow Error: 59.6 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-frac Error: 59.6 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: 9.2 bits

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

   8. Simplified Error: 9.2 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 5 regimes into one program.

4. Final simplification Error: 18.2 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.0802856481292018 \cdot 10^{+127}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\left(\log 1 + \log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \leq -1.8632042262810938 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 2.424827901233389 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{0.5}{\sqrt{\log 10}}} \cdot \frac{\log 1 + 2 \cdot \log im}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \leq 5.509693746587087 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log re\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))