Average Error: 37.2 → 0.5
Time: 34.7s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\sqrt[3]{{\left(\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))\right)}^{3}}} - 1)^*\right) \le -1.895622369577088 \cdot 10^{-08}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\sqrt[3]{{\left(\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))\right)}^{3}}} - 1)^*\right) \le 4.328522179700643 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.5
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 (* 2 (* (sin (/ eps 2)) (expm1 (cbrt (pow (log1p (cos (/ (fma 2 x eps) 2))) 3))))) < -1.895622369577088e-08

    1. Initial program 30.6

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -1.895622369577088e-08 < (* 2 (* (sin (/ eps 2)) (expm1 (cbrt (pow (log1p (cos (/ (fma 2 x eps) 2))) 3))))) < 4.328522179700643e-16

    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 simplify0.2

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

    6. Applied expm1-log1p-u0.2

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

    if 4.328522179700643e-16 < (* 2 (* (sin (/ eps 2)) (expm1 (cbrt (pow (log1p (cos (/ (fma 2 x eps) 2))) 3)))))

    1. Initial program 30.6

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

    3. Applied sin-sum0.8

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

Runtime

Time bar (total: 34.7s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))