math.cos on complex, imaginary part

Average Error: 43.7 → 0.5

Time: 10.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -0.014382893694163904:\\ \;\;\;\;0.5 \cdot \frac{\sin re}{e^{im}} + e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} + {im}^{5} \cdot \frac{-1}{60}\right)\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original	43.7
Target	0.3
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if im < -0.014382893694163904

   1. Initial program 4.4

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
   2. Using strategy rm

   3. Applied sub-neg 4.4

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

   4. Applied distribute-lft-in 4.6

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

   5. Simplified 4.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin re}{e^{im}}} + \left(0.5 \cdot \sin re\right) \cdot \left(-e^{im}\right)\]

   6. Simplified 4.4

      \[\leadsto 0.5 \cdot \frac{\sin re}{e^{im}} + \color{blue}{e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)}\]

   if -0.014382893694163904 < im

   1. Initial program 44.1

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

   2. Taylor expanded around 0 0.4

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

   3. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -0.014382893694163904:\\ \;\;\;\;0.5 \cdot \frac{\sin re}{e^{im}} + e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} + {im}^{5} \cdot \frac{-1}{60}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))