math.sin on complex, imaginary part

Average Error: 57.9 → 0.4

Time: 10.3s

Precision: binary64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;e^{0.0 - im} - e^{im} \le 3.655542065597 \cdot 10^{-4}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3} + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original	57.9
Target	0.3
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (exp (- 0.0 im)) (exp im)) < 3.655542065597e-4

   1. Initial program 58.5

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

   2. Taylor expanded around 0 0.4

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

   3. Simplified 0.4

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

   if 3.655542065597e-4 < (- (exp (- 0.0 im)) (exp im))

   1. Initial program 2.0

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

   2. Taylor expanded around inf 2.0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0.0 - im} - e^{im} \le 3.655542065597 \cdot 10^{-4}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3} + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (neg (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))