Average Error: 43.4 → 0.8
Time: 39.8s
Precision: 64
Internal Precision: 1344
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\left(\log \left(e^{(\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*}\right)\right) \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}{2} \cdot \sin y i\right))\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]

  3. Simplified0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{(\left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}}{2} \cdot \sin y i\right))\]
  4. Using strategy rm

  5. Applied add-log-exp0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\color{blue}{\left(\log \left(e^{(\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*}\right)\right)} \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}{2} \cdot \sin y i\right))\]

  6. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\left(\log \left(e^{(\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*}\right)\right) \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}{2} \cdot \sin y i\right))\]

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))