Average Error: 37.2 → 0.4
Time: 41.2s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le -2.2602076052664956 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le 3.409624133979621 \cdot 10^{-07}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (* 2 (* (* (sin (/ eps 2)) (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (cos (/ (+ x (+ eps x)) 2)) (cos (/ (+ x (+ eps x)) 2)))))) < -2.2602076052664956e-08

    1. Initial program 30.1

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

    3. Applied sin-sum0.5

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

    if -2.2602076052664956e-08 < (* 2 (* (* (sin (/ eps 2)) (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (cos (/ (+ x (+ eps x)) 2)) (cos (/ (+ x (+ eps x)) 2)))))) < 3.409624133979621e-07

    1. Initial program 44.9

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

    3. Applied diff-sin44.9

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

    4. Applied simplify0.4

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

    if 3.409624133979621e-07 < (* 2 (* (* (sin (/ eps 2)) (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (cos (/ (+ x (+ eps x)) 2)) (cos (/ (+ x (+ eps x)) 2))))))

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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: 41.2s)Debug logProfile

herbie shell --seed 2018201 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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