Average Error: 14.5 → 0.4
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\frac{\log_* (1 + (e^{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right)} - 1)^*) - \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)}{\cos a \cdot \cos b + \sin b \cdot \sin a}}\]

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 14.5

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

  3. Applied cos-sum0.3

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

  5. Applied flip--0.4

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

  7. Applied log1p-expm1-u0.4

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

  8. Final simplification0.4

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

Runtime

Time bar (total: 1.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.40.10.30%
herbie shell --seed 2018295 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))