Average Error: 37.1 → 0.5
Time: 35.4s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + \left(\sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*} \cdot \sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*}\right) \cdot \sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*})\right) \le -0.0001604010969912868:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + \left(\sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*} \cdot \sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*}\right) \cdot \sqrt[3]{(e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*})\right) \le 8.720141963895404 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.3
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)) (log1p (* (* (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2)))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2))))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2)))))))) < -0.0001604010969912868

    1. Initial program 30.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -0.0001604010969912868 < (* 2 (* (sin (/ eps 2)) (log1p (* (* (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2)))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2))))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2)))))))) < 8.720141963895404e-11

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    4. Applied simplify0.4

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

    6. Applied expm1-log1p-u0.4

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

    if 8.720141963895404e-11 < (* 2 (* (sin (/ eps 2)) (log1p (* (* (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2)))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2))))) (cbrt (expm1 (cos (/ (+ x (+ eps x)) 2))))))))

    1. Initial program 29.5

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

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

Runtime

Time bar (total: 35.4s)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +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)))