Average Error: 36.7 → 1.6
Time: 28.0s
Precision: 64
Internal Precision: 2432
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \cdot \sqrt[3]{(e^{\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))} - 1)^*}\right) \le -0.2079043307148191:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right)}\right) \cdot \sqrt[3]{(e^{\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))} - 1)^*}\right) \le 0.00029756214984644506:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{(2 \cdot x + \varepsilon)_*}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.3
Herbie1.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (* 2 (* (* (* (cbrt (cos (/ (fma 2 x eps) 2))) (sin (/ eps 2))) (cbrt (cos (/ (fma 2 x eps) 2)))) (cbrt (expm1 (log1p (cos (/ (fma 2 x eps) 2))))))) < -0.2079043307148191

    1. Initial program 28.1

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

    3. Applied sin-sum0.4

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

    if -0.2079043307148191 < (* 2 (* (* (* (cbrt (cos (/ (fma 2 x eps) 2))) (sin (/ eps 2))) (cbrt (cos (/ (fma 2 x eps) 2)))) (cbrt (expm1 (log1p (cos (/ (fma 2 x eps) 2))))))) < 0.00029756214984644506

    1. Initial program 43.5

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

    3. Applied diff-sin43.5

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

    4. Applied simplify2.5

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

    6. Applied expm1-log1p-u2.6

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

    if 0.00029756214984644506 < (* 2 (* (* (* (cbrt (cos (/ (fma 2 x eps) 2))) (sin (/ eps 2))) (cbrt (cos (/ (fma 2 x eps) 2)))) (cbrt (expm1 (log1p (cos (/ (fma 2 x eps) 2)))))))

    1. Initial program 29.1

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

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

Runtime

Time bar (total: 28.0s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +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)))