Average Error: 37.1 → 0.5
Time: 32.6s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1538196709498795 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\varepsilon \le 1.0850938345021913 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.6
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.1538196709498795e-08 or 1.0850938345021913e-08 < eps

    1. Initial program 30.7

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

    3. Applied sin-sum0.6

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

    if -1.1538196709498795e-08 < eps < 1.0850938345021913e-08

    1. Initial program 43.9

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

    3. Applied diff-sin43.9

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

    4. Applied simplify0.3

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

    6. Applied log1p-expm1-u0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)}\right)\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 32.6s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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)))