Average Error: 43.2 → 0.8
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(e^{-im} - e^{im}\right)} \le 0.45310857367425755:\\ \;\;\;\;\left(\left(-0.5\right) \cdot \sin re\right) \cdot \left(\left(\frac{1}{3} \cdot {im}^{3} + 2 \cdot im\right) + {im}^{5} \cdot \frac{1}{60}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sin re}{e^{im}} - e^{im} \cdot \sin re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.2
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp (log (- (exp (- im)) (exp im)))) < 0.45310857367425755

    1. Initial program 44.3

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

    2. Taylor expanded around 0 0.2

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

    if 0.45310857367425755 < (exp (log (- (exp (- im)) (exp im))))

    1. Initial program 17.3

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

    2. Initial simplification18.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(e^{-im} - e^{im}\right)} \le 0.45310857367425755:\\ \;\;\;\;\left(\left(-0.5\right) \cdot \sin re\right) \cdot \left(\left(\frac{1}{3} \cdot {im}^{3} + 2 \cdot im\right) + {im}^{5} \cdot \frac{1}{60}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sin re}{e^{im}} - e^{im} \cdot \sin re\right)\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018215 
(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))))