Average Error: 12.5 → 0.3
Time: 16.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;r \le -1.03009841534229370834857547250500417677 \cdot 10^{-28} \lor \neg \left(r \le 3.277587723233422995576698390125871087617 \cdot 10^{113}\right):\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{w \cdot \left(\left(w \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array}\]
\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 -1.03009841534229370834857547250500417677 \cdot 10^{-28} \lor \neg \left(r \le 3.277587723233422995576698390125871087617 \cdot 10^{113}\right):\\
\;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\\

\mathbf{else}:\\
\;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{w \cdot \left(\left(w \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\

\end{array}
double f(double v, double w, double r) {
        double r26320 = 3.0;
        double r26321 = 2.0;
        double r26322 = r;
        double r26323 = r26322 * r26322;
        double r26324 = r26321 / r26323;
        double r26325 = r26320 + r26324;
        double r26326 = 0.125;
        double r26327 = v;
        double r26328 = r26321 * r26327;
        double r26329 = r26320 - r26328;
        double r26330 = r26326 * r26329;
        double r26331 = w;
        double r26332 = r26331 * r26331;
        double r26333 = r26332 * r26322;
        double r26334 = r26333 * r26322;
        double r26335 = r26330 * r26334;
        double r26336 = 1.0;
        double r26337 = r26336 - r26327;
        double r26338 = r26335 / r26337;
        double r26339 = r26325 - r26338;
        double r26340 = 4.5;
        double r26341 = r26339 - r26340;
        return r26341;
}
↓
double f(double v, double w, double r) {
        double r26342 = r;
        double r26343 = -1.0300984153422937e-28;
        bool r26344 = r26342 <= r26343;
        double r26345 = 3.277587723233423e+113;
        bool r26346 = r26342 <= r26345;
        double r26347 = !r26346;
        bool r26348 = r26344 || r26347;
        double r26349 = 3.0;
        double r26350 = 2.0;
        double r26351 = r26350 / r26342;
        double r26352 = r26351 / r26342;
        double r26353 = r26349 + r26352;
        double r26354 = 0.125;
        double r26355 = v;
        double r26356 = r26350 * r26355;
        double r26357 = r26349 - r26356;
        double r26358 = r26354 * r26357;
        double r26359 = w;
        double r26360 = r26359 * r26342;
        double r26361 = r26359 * r26360;
        double r26362 = r26361 * r26342;
        double r26363 = 1.0;
        double r26364 = r26363 - r26355;
        double r26365 = r26362 / r26364;
        double r26366 = r26358 * r26365;
        double r26367 = r26353 - r26366;
        double r26368 = 4.5;
        double r26369 = r26367 - r26368;
        double r26370 = r26342 * r26342;
        double r26371 = r26350 / r26370;
        double r26372 = r26349 + r26371;
        double r26373 = r26360 * r26342;
        double r26374 = r26359 * r26373;
        double r26375 = r26374 / r26364;
        double r26376 = r26358 * r26375;
        double r26377 = r26372 - r26376;
        double r26378 = r26377 - r26368;
        double r26379 = r26348 ? r26369 : r26378;
        return r26379;
}

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 2 regimes

  2. if r < -1.0300984153422937e-28 or 3.277587723233423e+113 < r

    1. Initial program 16.7

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

    3. Applied *-un-lft-identity16.7

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

    4. Applied times-frac9.2

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

    5. Simplified9.2

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

    7. Applied associate-*l*0.4

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

    9. Applied associate-/r*0.4

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

    if -1.0300984153422937e-28 < r < 3.277587723233423e+113

    1. Initial program 9.2

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

    3. Applied *-un-lft-identity9.2

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

    4. Applied times-frac7.1

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

    5. Simplified7.1

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

    7. Applied associate-*l*3.7

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

    9. Applied associate-*l*0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;r \le -1.03009841534229370834857547250500417677 \cdot 10^{-28} \lor \neg \left(r \le 3.277587723233422995576698390125871087617 \cdot 10^{113}\right):\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{w \cdot \left(\left(w \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))