Average Error: 36.6 → 0.5
Time: 50.8s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le -9.286795330551009 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le 5.062108322672078 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target15.4
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 2 regimes

  2. if (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < -9.286795330551009e-08 or 5.062108322672078e-12 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))))

    1. Initial program 30.1

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

    3. Applied sin-sum0.6

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

    if -9.286795330551009e-08 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < 5.062108322672078e-12

    1. Initial program 43.9

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

    3. Applied diff-sin43.9

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

    4. Applied simplify0.3

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

Runtime

Time bar (total: 50.8s)Debug logProfile

herbie shell --seed 2018178 +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)))