Average Error: 37.1 → 0.5
Time: 32.3s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le -0.00015965096829055196:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \cos \varepsilon \cdot \sin x\\ \mathbf{elif}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le 3.097729881299127 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.7
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 (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < -0.00015965096829055196

    1. Initial program 29.8

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

    if -0.00015965096829055196 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x)) < 3.097729881299127e-07

    1. Initial program 44.5

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

    3. Applied diff-sin44.5

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

    4. Simplified0.5

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

    6. Applied expm1-log1p-u0.5

      \[\leadsto 2 \cdot \left(\color{blue}{(e^{\log_* (1 + \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

    if 3.097729881299127e-07 < (- (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))) (sin x))

    1. Initial program 29.7

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

    3. Applied sin-sum0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le -0.00015965096829055196:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \cos \varepsilon \cdot \sin x\\ \mathbf{elif}\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x \le 3.097729881299127 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right))} - 1)^*\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \end{array}\]

Runtime

Time bar (total: 32.3s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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