Input Error: 36.5b
Output Error: 4.8b
Time: 43.4s
Precision: 64b
Ground Truth: 128b
\[\sin \left(x + \varepsilon\right) - \sin x\]
⬇
\[\begin{cases} \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon\right)}^2}{\sin x \cdot \cos \varepsilon - \cos x \cdot \sin \varepsilon} - \sin x & \text{when } \varepsilon \le -8.336964433172839 \cdot 10^{-83} \\ \varepsilon - \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{2} & \text{when } \varepsilon \le 9.15602328414514 \cdot 10^{-96} \\ \log \left(e^{\sin x \cdot \cos \varepsilon}\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right) & \text{otherwise} \end{cases}\]

Error

Bits error versus x
Bits error versus eps

Derivation

    if eps < -8.336964433172839e-83

    1. Initial program 30.3b

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

    3. Applied sin-sum 6.6b

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

    5. Applied flip-+ 6.7b

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

    if -8.336964433172839e-83 < eps < 9.15602328414514e-96

    1. Initial program 48.2b

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

    2. Applied taylor 9.9b

      \[\leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]

    3. Taylor expanded around 0 9.9b

      \[\leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]

    4. Applied simplify 0.2b

      \[\leadsto \color{blue}{\varepsilon - \left(\left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{2}}\]

    if 9.15602328414514e-96 < eps

    1. Initial program 30.9b

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

    3. Applied sin-sum 7.2b

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

    4. Applied associate--l+ 7.2b

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

    6. Applied add-log-exp 7.6b

      \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
  1. Removed slow pow expressions

Runtime

Total time: 43.4s Debug log

herbie --seed '#(2301961629 3764887294 1644872900 1289297679 3024945287 3367809679)'
(FPCore (x eps)
  :name "NMSE example 3.3"
  
  :target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))(- (sin (+ x eps)) (sin x)))