Average Error: 0.6 → 1.0
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{v + 1} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{v - 1}\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  3. Applied difference-of-sqr-11.0

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

  4. Applied *-un-lft-identity1.0

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

  5. Applied times-frac1.0

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))