Average Error: 32.1 → 0.3
Time: 6.4s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\frac{{\log 10}^{-0.5}}{-\sqrt{\log 10}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{{\log 10}^{-0.5}}{-\sqrt{\log 10}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right)

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (* (/ (pow (log 10.0) -0.5) (- (sqrt (log 10.0)))) (- (log (hypot re im)))))
double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}
↓
double code(double re, double im) {
	return (pow(log(10.0), -0.5) / -sqrt(log(10.0))) * -log(hypot(re, im));
}

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. Initial program 32.1

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

  3. Applied egg-rr0.6

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

  4. Applied egg-rr0.6

    \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

    \[\leadsto \frac{{\log 10}^{-0.5}}{-\sqrt{\log 10}} \cdot \left(-\log \left(\mathsf{hypot}\left(re, im\right)\right)\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))