Average Error: 15.6 → 0.4
Time: 30.3s
Precision: 64
Internal Precision: 1344
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b\right)}^{3} \cdot \left(\sin a \cdot \left(\sin a \cdot \sin a\right)\right)}}\]

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 15.6

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

  3. Applied cos-sum0.3

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

  5. Applied add-cbrt-cube0.4

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

  6. Applied add-cbrt-cube0.4

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

  7. Applied cbrt-unprod0.4

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

  8. Simplified0.4

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

  10. Applied add-cbrt-cube0.4

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

  11. Applied add-cbrt-cube0.4

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

  12. Applied cbrt-unprod0.4

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

  13. Applied rem-cube-cbrt0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Runtime

Time bar (total: 30.3s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))