Average Error: 37.5 → 0.6
Time: 48.1s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} + \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\varepsilon \cdot \frac{1}{2}} - \sqrt[3]{\varepsilon \cdot \frac{1}{2}} \cdot \left(\frac{1}{72} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{51840} \cdot {\varepsilon}^{4}\right)\right)\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \le -1515782.0331371354:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;\left(\left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)} + \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right)}\right) \cdot \left(\sqrt[3]{\varepsilon \cdot \frac{1}{2}} - \sqrt[3]{\varepsilon \cdot \frac{1}{2}} \cdot \left(\frac{1}{72} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{51840} \cdot {\varepsilon}^{4}\right)\right)\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \le 3.1725461234134967 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.5
Target15.4
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* (* (+ (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (cbrt (sin (/ eps 2)))) (- (cbrt (* eps 1/2)) (* (cbrt (* eps 1/2)) (+ (* 1/72 (* eps eps)) (* 1/51840 (pow eps 4)))))) (cos (/ (+ x (+ eps x)) 2))) < -1515782.0331371354

    1. Initial program 30.0

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

    3. Applied sin-sum0.4

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

    4. Applied associate--l+0.4

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

    if -1515782.0331371354 < (* (* (* (+ (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (cbrt (sin (/ eps 2)))) (- (cbrt (* eps 1/2)) (* (cbrt (* eps 1/2)) (+ (* 1/72 (* eps eps)) (* 1/51840 (pow eps 4)))))) (cos (/ (+ x (+ eps x)) 2))) < 3.1725461234134967e-09

    1. Initial program 44.9

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

    3. Applied diff-sin44.9

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

    4. Applied simplify0.6

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

    if 3.1725461234134967e-09 < (* (* (* (+ (cbrt (sin (/ eps 2))) (cbrt (sin (/ eps 2)))) (cbrt (sin (/ eps 2)))) (- (cbrt (* eps 1/2)) (* (cbrt (* eps 1/2)) (+ (* 1/72 (* eps eps)) (* 1/51840 (pow eps 4)))))) (cos (/ (+ x (+ eps x)) 2)))

    1. Initial program 30.6

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

    3. Applied sin-sum0.6

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

Runtime

Time bar (total: 48.1s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))