Average Error: 15.1 → 0.5
Time: 30.2s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*)\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) \cdot \log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) + \log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) \cdot \left(\cos a \cdot \cos b\right)\right)\right)\right) \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 *-un-lft-identity0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

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

  9. Applied log1p-expm1-u0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)}}\]
  10. Using strategy rm

  11. Applied flip3--0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) \cdot \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) + \left(\cos a \cdot \cos b\right) \cdot \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)\right)}}}\]

  12. Applied associate-/r/0.5

    \[\leadsto r \cdot \color{blue}{\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) \cdot \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) + \left(\cos a \cdot \cos b\right) \cdot \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)\right)\right)\right)}\]

  13. Final simplification0.5

    \[\leadsto \left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*)\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) \cdot \log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) + \log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*) \cdot \left(\cos a \cdot \cos b\right)\right)\right)\right) \cdot r\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))