Average Error: 43.7 → 0.6
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \log \left(e^{e^{-im} - e^{im}}\right) \le -0.7470955502318206:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \frac{e^{-im} \cdot e^{-im} - e^{im} \cdot e^{im}}{e^{-im} + e^{im}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \left(-0.5\right)\right) \cdot (im \cdot \left((im \cdot \left(\frac{1}{3} \cdot im\right) + 2)_*\right) + \left({im}^{5} \cdot \frac{1}{60}\right))_*\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* 0.5 (sin re)) (log (exp (- (exp (- im)) (exp im))))) < -0.7470955502318206

    1. Initial program 4.1

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

    3. Applied flip--16.2

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

    if -0.7470955502318206 < (* (* 0.5 (sin re)) (log (exp (- (exp (- im)) (exp im)))))

    1. Initial program 43.9

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

    2. Taylor expanded around 0 0.6

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

    3. Applied simplify0.6

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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