Average Error: 43.7 → 31.1
Time: 27.2s
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.2539484306577847 \cdot 10^{+38}:\\ \;\;\;\;-1.0 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot (\left(\frac{1}{\sqrt{e^{im}}}\right) \cdot \left(\log_* (1 + (e^{\frac{\sin re}{\sqrt{e^{im}}}} - 1)^*)\right) + \left(\sin re \cdot \left(-e^{im}\right)\right))_*\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original43.7
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 re < 1.2539484306577847e+38

    1. Initial program 39.7

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

    2. Initial simplification39.8

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

    3. Taylor expanded around 0 23.6

      \[\leadsto \color{blue}{-1.0 \cdot \left(re \cdot im\right)}\]

    if 1.2539484306577847e+38 < re

    1. Initial program 58.3

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

    2. Initial simplification58.3

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

    4. Applied add-sqr-sqrt58.4

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

    5. Applied *-un-lft-identity58.4

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

    6. Applied times-frac58.4

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

    7. Applied fma-neg58.4

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

    9. Applied log1p-expm1-u58.7

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.2539484306577847 \cdot 10^{+38}:\\ \;\;\;\;-1.0 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot (\left(\frac{1}{\sqrt{e^{im}}}\right) \cdot \left(\log_* (1 + (e^{\frac{\sin re}{\sqrt{e^{im}}}} - 1)^*)\right) + \left(\sin re \cdot \left(-e^{im}\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 27.2s)Debug logProfile

herbie shell --seed 2018255 +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))))