Average Error: 47.3 → 2.6
Time: 20.3s
Precision: 64
Internal precision: 1408
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
⬇
\[\left(\left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right) + \frac{1}{3} \cdot {im}^3\right) \cdot \left(\left(-0.5\right) \cdot \sin re\right)\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 47.3

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

  2. Applied taylor 2.6

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right)\]

  3. Taylor expanded around 0 2.6

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{60} \cdot {im}^{5} + \left(2 \cdot im + \frac{1}{3} \cdot {im}^{3}\right)\right)\right)}\]

  4. Applied simplify 2.6

    \[\leadsto \color{blue}{\left(\left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right) + \frac{1}{3} \cdot {im}^3\right) \cdot \left(\left(-0.5\right) \cdot \sin re\right)}\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 20.3s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (re im)
  :name "Complex sine and cosine"
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))