Average Error: 40.2 → 1.0
Time: 45.0s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le -0.03945395694433318:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x \le 0.00029893097952776726:\\ \;\;\;\;\sin \left(\left(\varepsilon + x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]

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)) < -0.03945395694433318

    1. Initial program 21.2

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

    3. Applied cos-sum0.5

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

    if -0.03945395694433318 < (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)) < 0.00029893097952776726

    1. Initial program 47.5

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

    3. Applied diff-cos36.4

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

    4. Applied simplify1.4

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

    6. Applied add-cube-cbrt2.4

      \[\leadsto -2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right)} \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\]

    7. Applied associate-*l*2.4

      \[\leadsto -2 \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)}\]

    8. Taylor expanded around inf 2.4

      \[\leadsto -2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)}\right)\right)\]

    9. Applied simplify1.3

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

    if 0.00029893097952776726 < (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))

    1. Initial program 58.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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-0.8

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

Runtime

Time bar (total: 45.0s)Debug logProfile

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