Average Error: 36.7 → 0.6
Time: 36.1s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)\right) \le -2.3778336746087837 \cdot 10^{-15}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)\right) \le 6.5500521203891126 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.2
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 2 regimes

  2. if (* 2 (* (sin (/ eps 2)) (log1p (expm1 (cos (/ (+ x (+ eps x)) 2)))))) < -2.3778336746087837e-15 or 6.5500521203891126e-18 < (* 2 (* (sin (/ eps 2)) (log1p (expm1 (cos (/ (+ x (+ eps x)) 2))))))

    1. Initial program 29.6

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

    3. Applied sin-sum0.9

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

    4. Applied associate--l+0.9

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

    if -2.3778336746087837e-15 < (* 2 (* (sin (/ eps 2)) (log1p (expm1 (cos (/ (+ x (+ eps x)) 2)))))) < 6.5500521203891126e-18

    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 simplify0.1

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

    6. Applied log1p-expm1-u0.2

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

Runtime

Time bar (total: 36.1s)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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)))