Average Error: 36.4 → 0.7
Time: 39.1s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x \le -1.4498239245858109 \cdot 10^{-05}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x \le 6.724345379798816 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.4
Target14.9
Herbie0.7
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < -1.4498239245858109e-05

    1. Initial program 29.7

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

    if -1.4498239245858109e-05 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < 6.724345379798816e-133

    1. Initial program 48.1

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

    3. Applied diff-sin48.1

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

    4. Applied simplify0.4

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

    if 6.724345379798816e-133 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))

    1. Initial program 25.5

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

    3. Applied sin-sum1.3

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

Runtime

Time bar (total: 39.1s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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)))