Average Error: 37.3 → 0.6
Time: 28.2s
Precision: 64
Internal precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
⬇
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.6204066776528013 \cdot 10^{-10}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 6.909575705248122 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Comparison15.4
Herbie0.6
\[ 2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Split input into 3 regimes.

  2. if eps < -3.6204066776528013e-10

    1. Initial program 31.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied sin-sum 0.6

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

    if -3.6204066776528013e-10 < eps < 6.909575705248122e-21

    1. Initial program 44.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied diff-sin 44.3

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

    4. Applied simplify 0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)}\]

    if 6.909575705248122e-21 < eps

    1. Initial program 30.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied sin-sum 1.5

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

    4. Applied associate--l+ 1.5

      \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 28.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066372953 114334025 411438303 1288252006 2962405338 2829794477)'
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))