Average Error: 37.2 → 0.5
Time: 24.1s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right) \le -1.8956223695770888 \cdot 10^{-08}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right) \le 4.328522179700643 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

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

Derivation

  1. Split input into 3 regimes

  2. if (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ eps (+ x x)) 2))) (cbrt (cos (/ (+ eps (+ x x)) 2))))) (cbrt (cos (/ (+ eps (+ x x)) 2))))) < -1.8956223695770888e-08

    1. Initial program 30.6

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -1.8956223695770888e-08 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ eps (+ x x)) 2))) (cbrt (cos (/ (+ eps (+ x x)) 2))))) (cbrt (cos (/ (+ eps (+ x x)) 2))))) < 4.328522179700643e-16

    1. Initial program 44.4

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

    3. Applied diff-sin44.4

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

    4. Applied simplify0.2

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

    if 4.328522179700643e-16 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ eps (+ x x)) 2))) (cbrt (cos (/ (+ eps (+ x x)) 2))))) (cbrt (cos (/ (+ eps (+ x x)) 2)))))

    1. Initial program 30.6

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

    3. Applied sin-sum0.8

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

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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