Average Error: 0.6 → 1.0
Time: 33.9s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{{\left(v + 1\right)}^{3}} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 5}} \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v - 1}\right)\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  3. Applied difference-of-sqr-10.9

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

  4. Applied add-sqr-sqrt0.9

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

  5. Applied times-frac0.9

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

  7. Applied add-cbrt-cube1.0

    \[\leadsto \cos^{-1} \left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\color{blue}{\sqrt[3]{\left(\left(v + 1\right) \cdot \left(v + 1\right)\right) \cdot \left(v + 1\right)}}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - 1}\right)\]

  8. Applied add-cbrt-cube1.0

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\sqrt[3]{\left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}}{\sqrt[3]{\left(\left(v + 1\right) \cdot \left(v + 1\right)\right) \cdot \left(v + 1\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - 1}\right)\]

  9. Applied cbrt-undiv1.0

    \[\leadsto \cos^{-1} \left(\color{blue}{\sqrt[3]{\frac{\left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\left(\left(v + 1\right) \cdot \left(v + 1\right)\right) \cdot \left(v + 1\right)}}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - 1}\right)\]

  10. Simplified1.0

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{\sqrt{1 - \left(v \cdot v\right) \cdot 5} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{{\left(1 + v\right)}^{3}}}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - 1}\right)\]

  11. Final simplification1.0

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{{\left(v + 1\right)}^{3}} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 5}} \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v - 1}\right)\]

Runtime

Time bar (total: 33.9s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))