Average Error: 0.5 → 0.6
Time: 42.2s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\log_* (1 + (e^{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}} - 1)^*)\right)\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied log1p-expm1-u0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\log_* (1 + (e^{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}} - 1)^*)\right)}\]
  4. Using strategy rm

  5. Applied *-commutative0.6

    \[\leadsto \cos^{-1} \left(\log_* (1 + (e^{\frac{1 - \color{blue}{\left(v \cdot v\right) \cdot 5}}{v \cdot v - 1}} - 1)^*)\right)\]

  6. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\log_* (1 + (e^{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}} - 1)^*)\right)\]

Runtime

Time bar (total: 42.2s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))