Input Error: 37.1b
Output Error: 4.4b
Time: 52.8s
Precision: 64b
Ground Truth: 128b
\[\cos \left(x + \varepsilon\right) - \cos x\]
⬇
\[\begin{cases} \cos x \cdot \cos \varepsilon - \left(\log \left(e^{\sin x \cdot \sin \varepsilon}\right) + \cos x\right) & \text{when } \varepsilon \le -3.4500988550242373 \cdot 10^{-23} \\ \left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right) & \text{when } \varepsilon \le 0.4439134345824553 \\ \cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^3} + \cos x\right) & \text{otherwise} \end{cases}\]

Error

Bits error versus x
Bits error versus eps

Derivation

    if eps < -3.4500988550242373e-23

    1. Initial program 31.9b

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

    3. Applied cos-sum 4.0b

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

    4. Applied associate--l- 4.0b

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

    6. Applied add-log-exp 4.1b

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

    if -3.4500988550242373e-23 < eps < 0.4439134345824553

    1. Initial program 45.6b

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

    2. Applied taylor 7.0b

      \[\leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\varepsilon \cdot x + \frac{1}{2} \cdot {\varepsilon}^2\right)\]

    3. Taylor expanded around 0 7.0b

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\varepsilon \cdot x + \frac{1}{2} \cdot {\varepsilon}^2\right)}\]

    4. Applied simplify 7.0b

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]

    if 0.4439134345824553 < eps

    1. Initial program 30.3b

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

    3. Applied cos-sum 0.8b

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

    4. Applied associate--l- 0.8b

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

    6. Applied add-cbrt-cube 0.9b

      \[\leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^3}} + \cos x\right)\]
  1. Removed slow pow expressions

Runtime

Total time: 52.8s Debug log

herbie --seed '#(2353323701 2927288145 1346701523 2394139776 582644105 915241846)'
(FPCore (x eps)
  :name "NMSE problem 3.3.5"
  (- (cos (+ x eps)) (cos x)))