Average Error: 12.6 → 0.4
Time: 35.7s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;r \le -1.8424250382875904 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\\ \mathbf{elif}\;r \le 9.027945435515558 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \frac{\left(\left(r \cdot w\right) \cdot r\right) \cdot w}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{r \cdot r}} \cdot \left(\sqrt[3]{\frac{2}{r \cdot r}} \cdot \sqrt[3]{\frac{2}{r \cdot r}}\right) + 3\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if r < -1.8424250382875904e-47

    1. Initial program 12.5

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied associate-*l*7.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]
    4. Using strategy rm

    5. Applied *-un-lft-identity7.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

    6. Applied times-frac0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}}\right) - 4.5\]

    7. Simplified0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \left(v \cdot -2 + 3\right)\right)} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\]
    8. Using strategy rm

    9. Applied *-un-lft-identity0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

    10. Applied times-frac0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \color{blue}{\left(\frac{w \cdot \left(w \cdot r\right)}{1} \cdot \frac{r}{1 - v}\right)}\right) - 4.5\]

    11. Simplified0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot \frac{r}{1 - v}\right)\right) - 4.5\]
    12. Using strategy rm

    13. Applied associate-/r*0.6

      \[\leadsto \left(\left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\]

    if -1.8424250382875904e-47 < r < 9.027945435515558e+56

    1. Initial program 10.8

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied associate-*l*7.4

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]
    4. Using strategy rm

    5. Applied *-un-lft-identity7.4

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

    6. Applied times-frac4.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}}\right) - 4.5\]

    7. Simplified4.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \left(v \cdot -2 + 3\right)\right)} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\]
    8. Using strategy rm

    9. Applied associate-*l*0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \frac{\color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot r\right)}}{1 - v}\right) - 4.5\]

    if 9.027945435515558e+56 < r

    1. Initial program 16.9

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied associate-*l*8.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]
    4. Using strategy rm

    5. Applied *-un-lft-identity8.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

    6. Applied times-frac0.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}}\right) - 4.5\]

    7. Simplified0.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \left(v \cdot -2 + 3\right)\right)} \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\]
    8. Using strategy rm

    9. Applied *-un-lft-identity0.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{\color{blue}{1 \cdot \left(1 - v\right)}}\right) - 4.5\]

    10. Applied times-frac0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \color{blue}{\left(\frac{w \cdot \left(w \cdot r\right)}{1} \cdot \frac{r}{1 - v}\right)}\right) - 4.5\]

    11. Simplified0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot \frac{r}{1 - v}\right)\right) - 4.5\]
    12. Using strategy rm

    13. Applied add-cube-cbrt0.3

      \[\leadsto \left(\left(3 + \color{blue}{\left(\sqrt[3]{\frac{2}{r \cdot r}} \cdot \sqrt[3]{\frac{2}{r \cdot r}}\right) \cdot \sqrt[3]{\frac{2}{r \cdot r}}}\right) - \left(0.125 \cdot \left(v \cdot -2 + 3\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;r \le -1.8424250382875904 \cdot 10^{-47}:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\\ \mathbf{elif}\;r \le 9.027945435515558 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \frac{\left(\left(r \cdot w\right) \cdot r\right) \cdot w}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{r \cdot r}} \cdot \left(\sqrt[3]{\frac{2}{r \cdot r}} \cdot \sqrt[3]{\frac{2}{r \cdot r}}\right) + 3\right) - \left(\left(3 + v \cdot -2\right) \cdot 0.125\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) - 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))