Average Error: 30.6 → 0.4
Time: 1.6m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x \cdot x}} \cdot \sqrt[3]{\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x \cdot x}}\right) \cdot \sqrt[3]{\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x \cdot x}} \le 0.41414395715406016:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sin x \cdot \sin x}{(\left(\cos x\right) \cdot x + x)_*}\\ \mathbf{else}:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(\frac{1}{24} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))) (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x)))) (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x)))) < 0.41414395715406016

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied flip--1.2

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

    4. Applied simplify1.1

      \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\sin x \cdot \sin x}{1 + \cos x}}}{x \cdot x}\]

    7. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x}}\]

    8. Applied simplify0.6

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\sin x \cdot \sin x}{(\left(\cos x\right) \cdot x + x)_*}}\]

    if 0.41414395715406016 < (* (* (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))) (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x)))) (cbrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))))

    1. Initial program 60.9

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Applied simplify0.2

      \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(\frac{1}{24} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))