Average Error: 37.2 → 0.7
Time: 40.9s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right) \le -0.009223036685637842 \lor \neg \left(\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right) \le 0.0024644411836328532\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left((e^{\log_* (1 + \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^* \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.0
Herbie0.7
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x))) < -0.009223036685637842 or 0.0024644411836328532 < (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x)))

    1. Initial program 30.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -0.009223036685637842 < (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x))) < 0.0024644411836328532

    1. Initial program 44.0

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

    3. Applied diff-sin44.0

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

    4. Applied simplify0.9

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

    6. Applied expm1-log1p-u1.0

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

  4. Applied simplify0.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right) \le -0.009223036685637842 \lor \neg \left(\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right) \le 0.0024644411836328532\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left((e^{\log_* (1 + \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right))} - 1)^* \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}}\]

Runtime

Time bar (total: 40.9s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +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)))