Average Error: 30.7 → 9.3
Time: 36.9s
Precision: 64
Internal Precision: 1600
\[\frac{1}{\sin x} - \frac{1}{\sin \left(x - 10^{-06}\right)}\]
\[\frac{\sin x \cdot \cos \left( 10^{-06} \right) - \left(\cos x \cdot \sin \left( 10^{-06} \right) + \sin x\right)}{\left(\cos \left( 10^{-06} \right) \cdot \sin x - \cos x \cdot \sin \left( 10^{-06} \right)\right) \cdot \sin x}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

    \[\frac{1}{\sin x} - \frac{1}{\sin \left(x - 10^{-06}\right)}\]
  2. Using strategy rm
  3. Applied sub-neg30.7

    \[\leadsto \frac{1}{\sin x} - \frac{1}{\sin \color{blue}{\left(x + \left(-10^{-06}\right)\right)}}\]
  4. Applied sin-sum9.4

    \[\leadsto \frac{1}{\sin x} - \frac{1}{\color{blue}{\sin x \cdot \cos \left(-10^{-06}\right) + \cos x \cdot \sin \left(-10^{-06}\right)}}\]
  5. Applied simplify9.4

    \[\leadsto \frac{1}{\sin x} - \frac{1}{\color{blue}{\sin x \cdot \cos \left( 10^{-06} \right)} + \cos x \cdot \sin \left(-10^{-06}\right)}\]
  6. Using strategy rm
  7. Applied frac-sub9.3

    \[\leadsto \color{blue}{\frac{1 \cdot \left(\sin x \cdot \cos \left( 10^{-06} \right) + \cos x \cdot \sin \left(-10^{-06}\right)\right) - \sin x \cdot 1}{\sin x \cdot \left(\sin x \cdot \cos \left( 10^{-06} \right) + \cos x \cdot \sin \left(-10^{-06}\right)\right)}}\]
  8. Applied simplify9.3

    \[\leadsto \frac{\color{blue}{\sin x \cdot \cos \left( 10^{-06} \right) - \left(\cos x \cdot \sin \left( 10^{-06} \right) + \sin x\right)}}{\sin x \cdot \left(\sin x \cdot \cos \left( 10^{-06} \right) + \cos x \cdot \sin \left(-10^{-06}\right)\right)}\]
  9. Applied simplify9.3

    \[\leadsto \frac{\sin x \cdot \cos \left( 10^{-06} \right) - \left(\cos x \cdot \sin \left( 10^{-06} \right) + \sin x\right)}{\color{blue}{\left(\cos \left( 10^{-06} \right) \cdot \sin x - \cos x \cdot \sin \left( 10^{-06} \right)\right) \cdot \sin x}}\]

Runtime

Time bar (total: 36.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "1/sin(x)-1/(sin(x-0.000001))"
  (- (/ 1 (sin x)) (/ 1 (sin (- x 1e-06)))))