Average Error: 43.2 → 0.6
Time: 59.5s
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(e^{e^{-im} - e^{im}}\right) \cdot \left(0.5 \cdot \sin re\right) \le 0.004265090850955561:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot (\left(-im\right) \cdot \left((\left(\frac{1}{3} \cdot im\right) \cdot im + 2)_*\right) + \left({im}^{5} \cdot \left(-\frac{1}{60}\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot (1 \cdot \left(e^{-im}\right) + \left(\left(-\sqrt{e^{im}}\right) \cdot \sqrt{e^{im}}\right))_*\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original43.2
Target0.4
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))))) < 0.004265090850955561

    1. Initial program 43.5

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

    2. Taylor expanded around 0 0.6

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

    3. Simplified0.6

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

    if 0.004265090850955561 < (* (* 0.5 (sin re)) (log (exp (- (exp (- im)) (exp im)))))

    1. Initial program 5.7

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

    3. Applied add-sqr-sqrt6.0

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

    4. Applied *-un-lft-identity6.0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1 \cdot e^{-im}} - \sqrt{e^{im}} \cdot \sqrt{e^{im}}\right)\]

    5. Applied prod-diff5.9

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left((1 \cdot \left(e^{-im}\right) + \left(-\sqrt{e^{im}} \cdot \sqrt{e^{im}}\right))_* + (\left(-\sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{im}}\right) + \left(\sqrt{e^{im}} \cdot \sqrt{e^{im}}\right))_*\right)}\]

    6. Applied distribute-lft-in5.9

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot (1 \cdot \left(e^{-im}\right) + \left(-\sqrt{e^{im}} \cdot \sqrt{e^{im}}\right))_* + \left(0.5 \cdot \sin re\right) \cdot (\left(-\sqrt{e^{im}}\right) \cdot \left(\sqrt{e^{im}}\right) + \left(\sqrt{e^{im}} \cdot \sqrt{e^{im}}\right))_*}\]

    7. Simplified6.0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot (1 \cdot \left(e^{-im}\right) + \left(-\sqrt{e^{im}} \cdot \sqrt{e^{im}}\right))_* + \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{e^{-im} - e^{im}}\right) \cdot \left(0.5 \cdot \sin re\right) \le 0.004265090850955561:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot (\left(-im\right) \cdot \left((\left(\frac{1}{3} \cdot im\right) \cdot im + 2)_*\right) + \left({im}^{5} \cdot \left(-\frac{1}{60}\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot (1 \cdot \left(e^{-im}\right) + \left(\left(-\sqrt{e^{im}}\right) \cdot \sqrt{e^{im}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 59.5s)Debug logProfile

herbie shell --seed 2018215 +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))))