Average Error: 0.5 → 0.3
Time: 5.0m
Precision: 64
Internal Precision: 576
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{{\left(1 - \left(v \cdot v\right) \cdot 5\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{t}}{1 - v \cdot v}\]

Error

Bits error versus v

Bits error versus t

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Initial simplification0.4

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{t \cdot \pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{t \cdot \pi}}{\color{blue}{1 \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v}\]

  5. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5} \cdot \sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}\right) \cdot \sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}}{t \cdot \pi}}{1 \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]

  6. Applied times-frac0.4

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5} \cdot \sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{t} \cdot \frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}}{1 \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]

  7. Applied times-frac0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5} \cdot \sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{t}}{1} \cdot \frac{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v}\]

  8. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{{\left(1 - \left(v \cdot v\right) \cdot 5\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{t}} \cdot \frac{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]

  9. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{{\left(1 - \left(v \cdot v\right) \cdot 5\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{t}}{1 - v \cdot v}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))