Average Error: 37.7 → 1.3
Time: 1.1m
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.150392034090353 \cdot 10^{+38}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 8.180295516533624 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.7
Target14.9
Herbie1.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.150392034090353e+38 or 8.180295516533624e-13 < eps

    1. Initial program 29.9

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

    3. Applied sin-sum0.5

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

    if -1.150392034090353e+38 < eps < 8.180295516533624e-13

    1. Initial program 44.7

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

    3. Applied diff-sin44.7

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

    4. Applied simplify2.1

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

    6. Applied associate-*r*2.1

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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