Average Error: 36.9 → 0.5
Time: 29.3s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4672183838072302 \cdot 10^{-07} \lor \neg \left(\varepsilon \le 1.3025257202483418 \cdot 10^{-08}\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sqrt[3]{{\left(\cos \varepsilon \cdot \sin x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.1
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 < -1.4672183838072302e-07 or 1.3025257202483418e-08 < eps

    1. Initial program 29.6

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

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

    6. Applied add-cbrt-cube0.6

      \[\leadsto \sin x \cdot \color{blue}{\sqrt[3]{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]

    7. Applied add-cbrt-cube0.7

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]

    8. Applied cbrt-unprod0.6

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon\right)}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]

    9. Simplified0.6

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\cos \varepsilon \cdot \sin x\right)}^{3}}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]

    if -1.4672183838072302e-07 < eps < 1.3025257202483418e-08

    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. Simplified0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4672183838072302 \cdot 10^{-07} \lor \neg \left(\varepsilon \le 1.3025257202483418 \cdot 10^{-08}\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sqrt[3]{{\left(\cos \varepsilon \cdot \sin x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 29.3s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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