Average Error: 39.2 → 0.7
Time: 35.8s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right) \le -4.239488531214963 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right) \le 0.0002298365258066562:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* (sin (/ eps 2)) (sin (/ (fma 2 x eps) 2))) < -4.239488531214963e-09 or 0.0002298365258066562 < (* (sin (/ eps 2)) (sin (/ (fma 2 x eps) 2)))

    1. Initial program 30.2

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

    3. Applied cos-sum0.8

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

    if -4.239488531214963e-09 < (* (sin (/ eps 2)) (sin (/ (fma 2 x eps) 2))) < 0.0002298365258066562

    1. Initial program 48.8

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

    3. Applied diff-cos38.0

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

    4. Applied simplify0.5

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

Runtime

Time bar (total: 35.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))