Average Error: 39.7 → 0.7
Time: 17.3s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.737430187884421 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.016646715417618 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.737430187884421e-05

    1. Initial program 29.6

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

    2. Initial simplification29.6

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

    4. Applied cos-sum0.9

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

    if -4.737430187884421e-05 < eps < 4.016646715417618e-05

    1. Initial program 49.5

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

    2. Initial simplification49.5

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

    4. Applied diff-cos37.2

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

    5. Simplified0.4

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

    7. Applied associate-*r*0.4

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

    9. Applied log1p-expm1-u0.5

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

    if 4.016646715417618e-05 < eps

    1. Initial program 30.4

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

    2. Initial simplification30.4

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

    4. Applied cos-sum0.9

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

    5. Applied associate--l-0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.737430187884421 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.016646715417618 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \log_* (1 + (e^{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array}\]

Runtime

Time bar (total: 17.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes14.80.70.414.497.7%
herbie shell --seed 2018290 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))