Average Error: 37.0 → 0.5
Time: 38.9s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right)\right) \le -9.816138677263888 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}}\right)\right) \le 4.964896821683711 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target14.7
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)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (* (* (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2))))))) < -9.816138677263888e-08

    1. Initial program 29.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -9.816138677263888e-08 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (* (* (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2))))))) < 4.964896821683711e-09

    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 simplify0.3

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

    if 4.964896821683711e-09 < (* 2 (* (* (sin (/ eps 2)) (* (cbrt (cos (/ (+ x (+ eps x)) 2))) (cbrt (cos (/ (+ x (+ eps x)) 2))))) (* (* (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2)))) (cbrt (cbrt (cos (/ (+ x (+ eps x)) 2))))) (cbrt (cbrt (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: 38.9s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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