Average Error: 40.0 → 0.7
Time: 50.4s
Precision: 64
Internal Precision: 2368
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)} \cdot \sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\right) \cdot \sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)} \le -3.3298243624529644 \cdot 10^{-10}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{if}\;\left(\sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)} \cdot \sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\right) \cdot \sqrt[3]{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)} \le 3.7619897986149974 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (* (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))))) < -3.3298243624529644e-10

    1. Initial program 25.2

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.1

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

    if -3.3298243624529644e-10 < (* (* (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))))) < 3.7619897986149974e-06

    1. Initial program 49.8

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

    3. Applied diff-cos38.3

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

    4. Applied simplify0.4

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

    if 3.7619897986149974e-06 < (* (* (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2)))))) (cbrt (* -2 (* (sin (/ eps 2)) (sin (/ (+ x (+ eps x)) 2))))))

    1. Initial program 60.5

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

    3. Applied cos-sum1.2

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

Runtime

Time bar (total: 50.4s)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))