Average Error: 18.0 → 0.6
Time: 54.1s
Precision: 64
Internal Precision: 576
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;u \le -5.1932296497958944 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\sqrt[3]{\frac{t1 + u}{v}} \cdot \sqrt[3]{\frac{t1 + u}{v}}} \cdot \frac{\frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}}{\sqrt[3]{\frac{t1 + u}{v}}}\\ \mathbf{elif}\;u \le 2.3954696880265956 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\frac{-t1}{t1 + u}}{t1 + u}}{\frac{1}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{-t1}}{t1 + u}}{\sqrt[3]{\frac{t1 + u}{v}}} \cdot \left(\frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}}\right)\\ \end{array}\]

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. Split input into 3 regimes

  2. if u < -5.1932296497958944e+48

    1. Initial program 15.0

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

    2. Initial simplification1.4

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

    4. Applied add-cube-cbrt1.8

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

    5. Applied add-cube-cbrt1.9

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

    6. Applied add-cube-cbrt1.9

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

    7. Applied times-frac1.9

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

    8. Applied times-frac0.7

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

    if -5.1932296497958944e+48 < u < 2.3954696880265956e+51

    1. Initial program 20.4

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

    2. Initial simplification1.8

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

    4. Applied div-inv1.9

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

    5. Applied associate-/r*0.6

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

    if 2.3954696880265956e+51 < u

    1. Initial program 14.7

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

    2. Initial simplification1.1

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

    4. Applied add-cube-cbrt1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied add-cube-cbrt1.6

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

    7. Applied times-frac1.6

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

    8. Applied times-frac0.8

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

    9. Simplified0.8

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}}\right)} \cdot \frac{\frac{\sqrt[3]{-t1}}{t1 + u}}{\sqrt[3]{\frac{t1 + u}{v}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;u \le -5.1932296497958944 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\sqrt[3]{\frac{t1 + u}{v}} \cdot \sqrt[3]{\frac{t1 + u}{v}}} \cdot \frac{\frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}}{\sqrt[3]{\frac{t1 + u}{v}}}\\ \mathbf{elif}\;u \le 2.3954696880265956 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{\frac{-t1}{t1 + u}}{t1 + u}}{\frac{1}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{-t1}}{t1 + u}}{\sqrt[3]{\frac{t1 + u}{v}}} \cdot \left(\frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{\frac{t1 + u}{v}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 54.1s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))