Average Error: 37.3 → 0.6
Time: 54.2s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le -8.988080779252093 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le 4.230745930376923 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.2
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) < -8.988080779252093e-07 or 4.230745930376923e-21 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cos (/ (+ x (+ eps x)) 2)))))

    1. Initial program 30.2

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

    3. Applied sin-sum0.9

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

    if -8.988080779252093e-07 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) < 4.230745930376923e-21

    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 simplify0.2

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

Runtime

Time bar (total: 54.2s)Debug logProfile

herbie shell --seed 2018214 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))