Average Error: 17.7 → 1.6
Time: 33.2s
Precision: 64
Internal Precision: 320
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\left(\sqrt[3]{-v} \cdot \sqrt[3]{\frac{t1}{u + t1}}\right) \cdot \left(\sqrt[3]{-v} \cdot \sqrt[3]{\frac{t1}{u + t1}}\right)\right) \cdot \frac{\frac{\sqrt[3]{t1}}{\sqrt[3]{u + t1}}}{\frac{u + t1}{\sqrt[3]{-v}}}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.7

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]

  2. Initial simplification1.7

    \[\leadsto \frac{\frac{t1}{t1 + u}}{\frac{t1 + u}{-v}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt2.3

    \[\leadsto \frac{\frac{t1}{t1 + u}}{\frac{t1 + u}{\color{blue}{\left(\sqrt[3]{-v} \cdot \sqrt[3]{-v}\right) \cdot \sqrt[3]{-v}}}}\]

  5. Applied *-un-lft-identity2.3

    \[\leadsto \frac{\frac{t1}{t1 + u}}{\frac{\color{blue}{1 \cdot \left(t1 + u\right)}}{\left(\sqrt[3]{-v} \cdot \sqrt[3]{-v}\right) \cdot \sqrt[3]{-v}}}\]

  6. Applied times-frac2.3

    \[\leadsto \frac{\frac{t1}{t1 + u}}{\color{blue}{\frac{1}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}} \cdot \frac{t1 + u}{\sqrt[3]{-v}}}}\]

  7. Applied add-cube-cbrt2.4

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{t1}{t1 + u}} \cdot \sqrt[3]{\frac{t1}{t1 + u}}\right) \cdot \sqrt[3]{\frac{t1}{t1 + u}}}}{\frac{1}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}} \cdot \frac{t1 + u}{\sqrt[3]{-v}}}\]

  8. Applied times-frac1.6

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{t1}{t1 + u}} \cdot \sqrt[3]{\frac{t1}{t1 + u}}}{\frac{1}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}}} \cdot \frac{\sqrt[3]{\frac{t1}{t1 + u}}}{\frac{t1 + u}{\sqrt[3]{-v}}}}\]

  9. Simplified1.6

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{t1}{u + t1}} \cdot \sqrt[3]{-v}\right) \cdot \left(\sqrt[3]{\frac{t1}{u + t1}} \cdot \sqrt[3]{-v}\right)\right)} \cdot \frac{\sqrt[3]{\frac{t1}{t1 + u}}}{\frac{t1 + u}{\sqrt[3]{-v}}}\]
  10. Using strategy rm

  11. Applied cbrt-div1.6

    \[\leadsto \left(\left(\sqrt[3]{\frac{t1}{u + t1}} \cdot \sqrt[3]{-v}\right) \cdot \left(\sqrt[3]{\frac{t1}{u + t1}} \cdot \sqrt[3]{-v}\right)\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{t1}}{\sqrt[3]{t1 + u}}}}{\frac{t1 + u}{\sqrt[3]{-v}}}\]

  12. Final simplification1.6

    \[\leadsto \left(\left(\sqrt[3]{-v} \cdot \sqrt[3]{\frac{t1}{u + t1}}\right) \cdot \left(\sqrt[3]{-v} \cdot \sqrt[3]{\frac{t1}{u + t1}}\right)\right) \cdot \frac{\frac{\sqrt[3]{t1}}{\sqrt[3]{u + t1}}}{\frac{u + t1}{\sqrt[3]{-v}}}\]

Runtime

Time bar (total: 33.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.61.60.01.60%
herbie shell --seed 2018295 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))