Average Error: 0.5 → 0.5
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\log \left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\right)\]

Error

Bits error versus v

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied add-log-exp0.5

    \[\leadsto \frac{\pi}{2} - \color{blue}{\log \left(e^{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)}\]

  6. Applied add-log-exp0.5

    \[\leadsto \color{blue}{\log \left(e^{\frac{\pi}{2}}\right)} - \log \left(e^{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\]

  7. Applied diff-log0.5

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\right)}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))