math.cos on complex, imaginary part

Average Error: 43.9 → 0.7

Time: 8.2s

Precision: binary64

Math

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

↓

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

Error

Bits error versus re

Bits error versus im

Target

Original	43.9
Target	0.3
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.9

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

2. Taylor expanded around 0 0.7

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

3. Simplified 0.7

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

5. Applied distribute-lft-in 0.7

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

6. Simplified 0.7

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

7. Simplified 0.7

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

8. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (neg (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (neg im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (neg im)) (exp im))))