Average Error: 36.5 → 0.5
Time: 47.1s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le -0.0023139190919417196:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le 4.918498226073161 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\left(x \cdot 2 + \varepsilon\right) \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target14.8
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 2 regimes

  2. if (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < -0.0023139190919417196 or 4.918498226073161e-09 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))))

    1. Initial program 29.1

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

    3. Applied sin-sum0.5

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

    if -0.0023139190919417196 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < 4.918498226073161e-09

    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 simplify0.5

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

    6. Applied add-cbrt-cube0.7

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right)\]

    7. Applied simplify0.7

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\color{blue}{{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}^{3}}}\right)\]

    8. Taylor expanded around inf 0.7

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{{\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}}^{3}}\right)\]

    9. Applied simplify0.5

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\left(x \cdot 2 + \varepsilon\right) \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 47.1s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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