Average Error: 57.9 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \le 0.0047751775683070935:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left((\left(im \cdot \frac{1}{3}\right) \cdot im + 2)_* \cdot \left(-im\right)\right) + \left(\cos re \cdot \left(-0.5\right)\right) \cdot \left({im}^{5} \cdot \frac{1}{60}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re\right) \cdot 0.5\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original57.9
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))) < 0.0047751775683070935

    1. Initial program 58.4

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{(\left(-im\right) \cdot \left((\left(\frac{1}{3} \cdot im\right) \cdot im + 2)_*\right) + \left({im}^{5} \cdot \left(-\frac{1}{60}\right)\right))_*}\]
    4. Using strategy rm

    5. Applied fma-udef0.5

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

    6. Applied distribute-lft-in0.5

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

    if 0.0047751775683070935 < (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im)))

    1. Initial program 1.2

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

    2. Initial simplification1.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \le 0.0047751775683070935:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left((\left(im \cdot \frac{1}{3}\right) \cdot im + 2)_* \cdot \left(-im\right)\right) + \left(\cos re \cdot \left(-0.5\right)\right) \cdot \left({im}^{5} \cdot \frac{1}{60}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re\right) \cdot 0.5\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))