Average Error: 43.4 → 31.1
Time: 31.5s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le 4.2230976061658064 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}} \cdot \sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}}\right) \cdot \sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}}\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie31.1
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < 4.2230976061658064e-14

    1. Initial program 44.1

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

    2. Simplified44.1

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

    3. Taylor expanded around 0 31.5

      \[\leadsto \color{blue}{\left(-2 \cdot \left(re \cdot im\right)\right)} \cdot 0.5\]

    if 4.2230976061658064e-14 < im

    1. Initial program 15.8

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

    2. Simplified16.7

      \[\leadsto \color{blue}{\left(\frac{\sin re}{e^{im}} - e^{im} \cdot \sin re\right) \cdot 0.5}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt16.9

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le 4.2230976061658064 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}} \cdot \sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}}\right) \cdot \sqrt[3]{\frac{\sin re}{e^{im}} - \sin re \cdot e^{im}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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))))