Average Error: 12.6 → 4.6
Time: 19.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(r \cdot \left(r \cdot w\right)\right) \cdot w}}\right) - 4.5\]
\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
↓
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(r \cdot \left(r \cdot w\right)\right) \cdot w}}\right) - 4.5
double f(double v, double w, double r) {
        double r27059 = 3.0;
        double r27060 = 2.0;
        double r27061 = r;
        double r27062 = r27061 * r27061;
        double r27063 = r27060 / r27062;
        double r27064 = r27059 + r27063;
        double r27065 = 0.125;
        double r27066 = v;
        double r27067 = r27060 * r27066;
        double r27068 = r27059 - r27067;
        double r27069 = r27065 * r27068;
        double r27070 = w;
        double r27071 = r27070 * r27070;
        double r27072 = r27071 * r27061;
        double r27073 = r27072 * r27061;
        double r27074 = r27069 * r27073;
        double r27075 = 1.0;
        double r27076 = r27075 - r27066;
        double r27077 = r27074 / r27076;
        double r27078 = r27064 - r27077;
        double r27079 = 4.5;
        double r27080 = r27078 - r27079;
        return r27080;
}
↓
double f(double v, double w, double r) {
        double r27081 = 3.0;
        double r27082 = 2.0;
        double r27083 = r;
        double r27084 = r27083 * r27083;
        double r27085 = r27082 / r27084;
        double r27086 = r27081 + r27085;
        double r27087 = 0.125;
        double r27088 = v;
        double r27089 = r27082 * r27088;
        double r27090 = r27081 - r27089;
        double r27091 = r27087 * r27090;
        double r27092 = 1.0;
        double r27093 = r27092 - r27088;
        double r27094 = w;
        double r27095 = r27083 * r27094;
        double r27096 = r27083 * r27095;
        double r27097 = r27096 * r27094;
        double r27098 = r27093 / r27097;
        double r27099 = r27091 / r27098;
        double r27100 = r27086 - r27099;
        double r27101 = 4.5;
        double r27102 = r27100 - r27101;
        return r27102;
}

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 < -93.93746802656042 or 1.2139398459933188e-49 < r

    1. Initial program 13.0

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

    3. Applied associate-*l*7.8

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

    5. Applied associate-/l*0.5

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

    7. Applied *-un-lft-identity0.5

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

    8. Applied times-frac0.6

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

    9. Applied times-frac0.6

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

    10. Simplified0.6

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

    12. Applied associate-/r/0.5

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

    13. Applied associate-*r*0.4

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

    if -93.93746802656042 < r < 1.2139398459933188e-49

    1. Initial program 11.8

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

    3. Applied associate-*l*7.8

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

    5. Applied associate-/l*5.5

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

    6. Taylor expanded around 0 24.6

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

    7. Simplified0.3

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

  4. Final simplification4.6

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

Reproduce

herbie shell --seed 2019291 
(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))