Average Error: 37.1 → 0.5
Time: 20.7s
Precision: 64
Internal Precision: 2368
\[\sin x - \sin \left(x + y\right)\]
\[\log \left({\left(e^{\sin x}\right)}^{\left(1 - \cos y\right)}\right) - \cos x \cdot \sin y\]

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 37.1

    \[\sin x - \sin \left(x + y\right)\]
  2. Using strategy rm
  3. Applied sin-sum22.2

    \[\leadsto \sin x - \color{blue}{\left(\sin x \cdot \cos y + \cos x \cdot \sin y\right)}\]
  4. Applied associate--r+0.4

    \[\leadsto \color{blue}{\left(\sin x - \sin x \cdot \cos y\right) - \cos x \cdot \sin y}\]
  5. Using strategy rm
  6. Applied add-log-exp14.9

    \[\leadsto \left(\sin x - \color{blue}{\log \left(e^{\sin x \cdot \cos y}\right)}\right) - \cos x \cdot \sin y\]
  7. Applied add-log-exp0.5

    \[\leadsto \left(\color{blue}{\log \left(e^{\sin x}\right)} - \log \left(e^{\sin x \cdot \cos y}\right)\right) - \cos x \cdot \sin y\]
  8. Applied diff-log0.5

    \[\leadsto \color{blue}{\log \left(\frac{e^{\sin x}}{e^{\sin x \cdot \cos y}}\right)} - \cos x \cdot \sin y\]
  9. Simplified0.5

    \[\leadsto \log \color{blue}{\left({\left(e^{\sin x}\right)}^{\left(1 - \cos y\right)}\right)} - \cos x \cdot \sin y\]
  10. Final simplification0.5

    \[\leadsto \log \left({\left(e^{\sin x}\right)}^{\left(1 - \cos y\right)}\right) - \cos x \cdot \sin y\]

Runtime

Time bar (total: 20.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y)
  :name "sin(x)-sin(x+y)"
  (- (sin x) (sin (+ x y))))