math.log/1 on complex, real part

Average Error: 31.6 → 18.3

Time: 2.2s

Precision: binary64

Math

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↓

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Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -6.66181232792173805e32

   1. Initial program 42.3

      \[\]

   2. Taylor expanded around -inf 11.8

      \[\leadsto \]

   3. Simplified 11.8

      \[\leadsto \]

   if -6.66181232792173805e32 < re < -4.85032597873741902e-160 or 8.69468551922083389e-226 < re < 2.757432490321271e106

   1. Initial program 18.2

      \[\]

   if -4.85032597873741902e-160 < re < 8.69468551922083389e-226

   1. Initial program 30.9

      \[\]

   2. Taylor expanded around 0 34.5

      \[\leadsto \]

   if 2.757432490321271e106 < re

   1. Initial program 52.6

      \[\]

   2. Taylor expanded around inf 9.2

      \[\leadsto \]
3. Recombined 4 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))