Average Error: 40.0 → 0.9
Time: 39.4s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right) \le -0.0011816087204703648:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;-2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right) \le 0.001002838732376614:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (* -2 (* (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (* (cbrt (sin (/ eps 2))) (sin (/ (+ x (+ eps x)) 2))))) < -0.0011816087204703648

    1. Initial program 26.1

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

    3. Applied cos-sum0.7

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

    if -0.0011816087204703648 < (* -2 (* (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (* (cbrt (sin (/ eps 2))) (sin (/ (+ x (+ eps x)) 2))))) < 0.001002838732376614

    1. Initial program 48.8

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

    3. Applied diff-cos36.9

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

    4. Applied simplify0.9

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

    if 0.001002838732376614 < (* -2 (* (* (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (* (cbrt (sin (/ eps 2))) (sin (/ (+ x (+ eps x)) 2)))))

    1. Initial program 60.8

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

    3. Applied cos-sum1.2

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

    4. Applied associate--l-1.2

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

Runtime

Time bar (total: 39.4s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))