Average Error: 39.3 → 0.8
Time: 48.5s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le -0.008103264256913323:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le 0.0031162088837781536:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)) < -0.008103264256913323

    1. Initial program 21.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.008103264256913323 < (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)) < 0.0031162088837781536

    1. Initial program 47.8

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

    3. Applied diff-cos36.5

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

    4. Applied simplify0.9

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

    6. Applied expm1-log1p-u1.0

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

    if 0.0031162088837781536 < (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))

    1. Initial program 58.5

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

    5. Applied simplify0.8

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

Runtime

Time bar (total: 48.5s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))