Average Error: 39.6 → 0.8
Time: 37.1s
Precision: 64
Internal Precision: 2432
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot \left(\left((e^{\log_* (1 + \sqrt[3]{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)})} - 1)^* \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}^{3}}\right) \le -0.003002798055434408:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \mathbf{if}\;-2 \cdot \left(\left((e^{\log_* (1 + \sqrt[3]{\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)})} - 1)^* \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)}^{3}}\right) \le 1.7496434683797861 \cdot 10^{-06}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) \cdot \left(\sin x\right) + \left(\cos x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* -2 (* (* (expm1 (log1p (cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2))) (cbrt (pow (cbrt (sin (/ (+ x (+ eps x)) 2))) 3)))) < -0.003002798055434408 or 1.7496434683797861e-06 < (* -2 (* (* (expm1 (log1p (cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2))) (cbrt (pow (cbrt (sin (/ (+ x (+ eps x)) 2))) 3))))

    1. Initial program 30.1

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

    3. Applied cos-sum0.7

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

    4. Applied associate--l-0.8

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

    5. Applied simplify0.7

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

    if -0.003002798055434408 < (* -2 (* (* (expm1 (log1p (cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2))) (cbrt (pow (cbrt (sin (/ (+ x (+ eps x)) 2))) 3)))) < 1.7496434683797861e-06

    1. Initial program 48.9

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

    3. Applied diff-cos36.6

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

    4. Applied simplify0.9

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))