Average Error: 0.6 → 0.7
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1} \cdot \sqrt[3]{v \cdot v - 1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt[3]{v \cdot v - 1}}\right)\]

Error

Bits error versus v

Try it out

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 add-cube-cbrt0.6

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

  4. Applied add-cube-cbrt0.7

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

  5. Applied times-frac0.7

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))