Average Error: 57.9 → 29.2
Time: 40.7s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.488568709968071:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;re \le 17220051194985.008:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(\frac{-1}{3} \cdot im\right) + \left(-2 + re \cdot re\right)\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot \frac{1}{e^{im}} - \cos re \cdot e^{im}\right) \cdot 0.5\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
Target0.3
Herbie29.2
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -4.488568709968071

    1. Initial program 57.9

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

    if -4.488568709968071 < re < 17220051194985.008

    1. Initial program 57.8

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

    2. Initial simplification57.9

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

    3. Taylor expanded around 0 2.4

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

    4. Simplified2.4

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

    if 17220051194985.008 < re

    1. Initial program 58.0

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

    2. Initial simplification58.0

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

    4. Applied div-inv58.0

      \[\leadsto 0.5 \cdot \left(\color{blue}{\cos re \cdot \frac{1}{e^{im}}} - e^{im} \cdot \cos re\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification29.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.488568709968071:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;re \le 17220051194985.008:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(\frac{-1}{3} \cdot im\right) + \left(-2 + re \cdot re\right)\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot \frac{1}{e^{im}} - \cos re \cdot e^{im}\right) \cdot 0.5\\ \end{array}\]

Runtime

Time bar (total: 40.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.329.228.23.068.2%
herbie shell --seed 2018352 
(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))))