Average Error: 57.9 → 1.0
Time: 45.0s
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(im \cdot \cos re\right) \cdot \left(-1.0\right) + \left(0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(-im\right) \cdot \cos re\right)\]

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

Original57.9
Target0.2
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Taylor expanded around inf 1.0

    \[\leadsto \color{blue}{-\left(1.0 \cdot \left(im \cdot \cos re\right) + 0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right)}\]

  5. Simplified1.0

    \[\leadsto \color{blue}{\left(\cos re \cdot \left(-im\right)\right) \cdot (0.16666666666666666 \cdot \left(im \cdot im\right) + 1.0)_*}\]
  6. Using strategy rm

  7. Applied fma-udef1.0

    \[\leadsto \left(\cos re \cdot \left(-im\right)\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(im \cdot im\right) + 1.0\right)}\]

  8. Applied distribute-lft-in1.0

    \[\leadsto \color{blue}{\left(\cos re \cdot \left(-im\right)\right) \cdot \left(0.16666666666666666 \cdot \left(im \cdot im\right)\right) + \left(\cos re \cdot \left(-im\right)\right) \cdot 1.0}\]

  9. Final simplification1.0

    \[\leadsto \left(im \cdot \cos re\right) \cdot \left(-1.0\right) + \left(0.16666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(\left(-im\right) \cdot \cos re\right)\]

Runtime

Time bar (total: 45.0s)Debug logProfile

herbie shell --seed 2018234 +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))))