Average Error: 37.2 → 0.4
Time: 2.5m
Precision: 64
Ground Truth: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
⬇
\[\left(\cos \varepsilon \cdot \sin x - \sin x\right) + \sin \varepsilon \cdot \cos x\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Comparison14.7
Herbie0.4
\[ 2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.2

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

  3. Applied sin-sum 22.3

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

  5. Applied add-cbrt-cube 23.0

    \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^3}} + \cos x \cdot \sin \varepsilon\right) - \sin x\]

  6. Applied taylor 23.0

    \[\leadsto \left(\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^3} + \cos x \cdot \sin \varepsilon\right) - \sin x\]

  7. Taylor expanded around 0 23.0

    \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\sin x \cdot \cos \varepsilon\right)}^3}} + \cos x \cdot \sin \varepsilon\right) - \sin x\]

  8. Applied simplify 0.4

    \[\leadsto \color{blue}{\left(\cos \varepsilon \cdot \sin x - \sin x\right) + \sin \varepsilon \cdot \cos x}\]
  9. Removed slow pow expressions

Runtime

Total time: 2.5m Debug log

Please include this information when filing a bug report:

herbie --seed '#(207320870 3046384572 2871534793 3373030187 1410669607 3523544200)'
(FPCore (x eps)
  :name "NMSE example 3.3"

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

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