math.log10 on complex, real part

Average Error: 31.9 → 18.1

Time: 6.8s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -7.2724683391263535e32

   1. Initial program 42.5

      \[\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 42.5

      \[\leadsto \]

   4. Applied pow1 42.5

      \[\leadsto \]

   5. Applied log-pow 42.5

      \[\leadsto \]

   6. Applied times-frac 42.5

      \[\leadsto \]
   7. Using strategy rm

   8. Applied add-log-exp 42.5

      \[\leadsto \]

   9. Simplified 42.4

      \[\leadsto \]

   10. Taylor expanded around -inf 12.0

       \[\leadsto \]

   11. Simplified 12.0

       \[\leadsto \]

   if -7.2724683391263535e32 < re < -1.6831834820475888e-179

   1. Initial program 17.9

      \[\]
   2. Using strategy rm

   3. Applied pow1/2 17.9

      \[\leadsto \]

   4. Applied log-pow 17.9

      \[\leadsto \]

   5. Applied associate-/l* 17.9

      \[\leadsto \]
   6. Using strategy rm

   7. Applied pow1 17.9

      \[\leadsto \]

   8. Applied log-pow 17.9

      \[\leadsto \]

   9. Applied add-cube-cbrt 18.5

      \[\leadsto \]

   10. Applied times-frac 18.5

       \[\leadsto \]

   11. Applied add-cube-cbrt 17.9

       \[\leadsto \]

   12. Applied times-frac 17.8

       \[\leadsto \]

   13. Simplified 17.8

       \[\leadsto \]

   14. Simplified 17.7

       \[\leadsto \]

   if -1.6831834820475888e-179 < re < -1.8532320640870052e-259

   1. Initial program 33.2

      \[\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 33.2

      \[\leadsto \]

   4. Applied pow1 33.2

      \[\leadsto \]

   5. Applied log-pow 33.2

      \[\leadsto \]

   6. Applied times-frac 33.2

      \[\leadsto \]
   7. Using strategy rm

   8. Applied add-log-exp 33.2

      \[\leadsto \]

   9. Simplified 33.1

      \[\leadsto \]

   10. Taylor expanded around 0 37.1

       \[\leadsto \]

   if -1.8532320640870052e-259 < re < 7.87144532982115844e108

   1. Initial program 22.6

      \[\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 22.6

      \[\leadsto \]

   4. Applied pow1 22.6

      \[\leadsto \]

   5. Applied log-pow 22.6

      \[\leadsto \]

   6. Applied times-frac 22.6

      \[\leadsto \]
   7. Using strategy rm

   8. Applied add-log-exp 22.6

      \[\leadsto \]

   9. Simplified 22.4

      \[\leadsto \]

   if 7.87144532982115844e108 < re

   1. Initial program 53.3

      \[\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 53.3

      \[\leadsto \]

   4. Applied pow1 53.3

      \[\leadsto \]

   5. Applied log-pow 53.3

      \[\leadsto \]

   6. Applied times-frac 53.3

      \[\leadsto \]
   7. Using strategy rm

   8. Applied add-log-exp 53.3

      \[\leadsto \]

   9. Simplified 53.3

      \[\leadsto \]

   10. Taylor expanded around inf 9.4

       \[\leadsto \]
3. Recombined 5 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))