Average Error: 36.6 → 0.4
Time: 30.5s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[(\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\left(\cos \varepsilon - 1\right) \cdot \sin x\right))_*\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.6
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.6

    \[\sin \left(x + \varepsilon\right) - \sin x\]

  2. Initial simplification36.6

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

  4. Applied sin-sum21.4

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

  5. Applied associate--l+0.4

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

  7. Applied *-un-lft-identity0.4

    \[\leadsto \sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right)\]

  8. Applied distribute-rgt-out--0.4

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(\cos \varepsilon - 1\right)}\]
  9. Using strategy rm

  10. Applied fma-def0.4

    \[\leadsto \color{blue}{(\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right))_*}\]

  11. Final simplification0.4

    \[\leadsto (\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\left(\cos \varepsilon - 1\right) \cdot \sin x\right))_*\]

Runtime

Time bar (total: 30.5s)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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