Average Error: 15.1 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b \cdot r}{\frac{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) \cdot \log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)}{\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*) + \cos b \cdot \cos a}}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

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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 log1p-expm1-u0.3

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

  7. Applied flip--0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Final simplification0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))