Average Error: 37.1 → 0.5
Time: 37.7s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \le -1.3580636772774767 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \le 2.9533449888865917 \cdot 10^{-05}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.3
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))) < -1.3580636772774767e-08

    1. Initial program 30.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -1.3580636772774767e-08 < (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x))) < 2.9533449888865917e-05

    1. Initial program 45.1

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

    3. Applied diff-sin45.1

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

    4. Applied simplify0.4

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

    6. Applied log1p-expm1-u0.5

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

    if 2.9533449888865917e-05 < (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (sin x)))

    1. Initial program 29.4

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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

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

Runtime

Time bar (total: 37.7s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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)))