Average Error: 31.8 → 18.1
Time: 5.5s
Precision: binary64
\[\]
↓
\[\]
↓
double code(double re, double im) {
	return ((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) / ((double) log(10.0))));
}

↓
double code(double re, double im) {
	double VAR;
	if ((re <= -9.390785678354852e+30)) {
		VAR = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) (((double) log(1.0)) - ((double) (2.0 * ((double) log(((double) (-1.0 / re)))))))) * ((double) sqrt(((double) (1.0 / ((double) log(10.0))))))))));
	} else {
		double VAR_1;
		if ((re <= -1.049508672528826e-189)) {
			VAR_1 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), ((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))));
		} else {
			double VAR_2;
			if ((re <= 1.3384531643325107e-226)) {
				VAR_2 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) * ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) log(im))))))))));
			} else {
				double VAR_3;
				if ((re <= 3.915439100569453e+101)) {
					VAR_3 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) log(((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), ((double) (1.0 / ((double) sqrt(((double) log(10.0))))))))))));
				} else {
					VAR_3 = ((double) (((double) (0.5 / ((double) sqrt(((double) log(10.0)))))) * ((double) (((double) sqrt(((double) (1.0 / ((double) log(10.0)))))) * ((double) (((double) log(1.0)) - ((double) (((double) log(re)) * -2.0))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 4 regimes
  2. if  re  < -9.39078567835485203e30
    1. Initial program 43.1
      \[\]
    2. Using strategy rm
    3. Applied add-sqr-sqrt43.1
      \[\leadsto \]
    4. Applied pow1/243.1
      \[\leadsto \]
    5. Applied log-pow43.1
      \[\leadsto \]
    6. Applied times-frac43.1
      \[\leadsto \]
    7. Taylor expanded around -inf 12.8
      \[\leadsto \]
    if -9.39078567835485203e30 <  re  < -1.0495086725288259e-189 or 1.3384531643325107e-226 <  re  < 3.9154391005694529e101
    1. Initial program 19.3
      \[\]
    2. Using strategy rm
    3. Applied add-sqr-sqrt19.3
      \[\leadsto \]
    4. Applied pow1/219.3
      \[\leadsto \]
    5. Applied log-pow19.3
      \[\leadsto \]
    6. Applied times-frac19.2
      \[\leadsto \]
    7. Using strategy rm
    8. Applied add-log-exp19.2
      \[\leadsto \]
    9. Simplified19.0
      \[\leadsto \]
    if -1.0495086725288259e-189 <  re  < 1.3384531643325107e-226
    1. Initial program 30.0
      \[\]
    2. Using strategy rm
    3. Applied add-sqr-sqrt30.0
      \[\leadsto \]
    4. Applied pow1/230.0
      \[\leadsto \]
    5. Applied log-pow30.0
      \[\leadsto \]
    6. Applied times-frac29.9
      \[\leadsto \]
    7. Taylor expanded around 0 33.3
      \[\leadsto \]
    8. Simplified33.3
      \[\leadsto \]
    if 3.9154391005694529e101 <  re 
    1. Initial program 52.0
      \[\]
    2. Using strategy rm
    3. Applied add-sqr-sqrt52.0
      \[\leadsto \]
    4. Applied pow1/252.0
      \[\leadsto \]
    5. Applied log-pow52.0
      \[\leadsto \]
    6. Applied times-frac52.0
      \[\leadsto \]
    7. Taylor expanded around inf 8.7
      \[\leadsto \]
    8. Simplified8.7
      \[\leadsto \]
  3. Recombined 4 regimes into one program.
  4. Final simplification18.1
    \[\leadsto \]
Reproduce
herbie shell --seed 2020179 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))