Average Error: 40.2 → 1.0
Time: 40.5s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le -0.03273608516386127:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_* \le 0.0005978619590555345:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left((2 \cdot x + \varepsilon)_* \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < -0.03273608516386127 or 0.0005978619590555345 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))

    1. Initial program 31.5

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

    3. Applied cos-sum0.6

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

    if -0.03273608516386127 < (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))) < 0.0005978619590555345

    1. Initial program 47.8

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

    3. Applied diff-cos36.6

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

    4. Applied simplify1.3

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

    5. Taylor expanded around inf 1.3

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

    6. Applied simplify1.3

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

Runtime

Time bar (total: 40.5s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))