Average Error: 14.8 → 0.4
Time: 31.4s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{6}}}}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]

  2. Initial simplification14.8

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
  3. Using strategy rm

  4. Applied cos-sum0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \color{blue}{\sqrt[3]{\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)}}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sqrt[3]{\left(\left(\sin b \cdot \sin a\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)}}\right) \cdot \left(\sin b \cdot \sin a\right)}}\]

  9. Applied add-cbrt-cube0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sqrt[3]{\left(\color{blue}{\sqrt[3]{\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)}} \cdot \sqrt[3]{\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)}\right) \cdot \left(\sin b \cdot \sin a\right)}}\]

  10. Applied cbrt-unprod0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sqrt[3]{\color{blue}{\sqrt[3]{\left(\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) \cdot \left(\sin b \cdot \sin a\right)\right)}} \cdot \left(\sin b \cdot \sin a\right)}}\]

  11. Simplified0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos b \cdot \cos a - \sqrt[3]{\sqrt[3]{\color{blue}{{\left(\sin a \cdot \sin b\right)}^{6}}} \cdot \left(\sin b \cdot \sin a\right)}}\]

  12. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{6}}}}\]

Runtime

Time bar (total: 31.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.40.10.30%
herbie shell --seed 2018290 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))