Average Error: 12.8 → 0.4
Time: 22.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \left(\frac{2}{r \cdot r} - \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|r \cdot w\right|\right)}^{2}} \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|r \cdot w\right|\right)}^{2}}\right)\right) - 4.5\]
\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(3 + \left(\frac{2}{r \cdot r} - \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|r \cdot w\right|\right)}^{2}} \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|r \cdot w\right|\right)}^{2}}\right)\right) - 4.5
double f(double v, double w, double r) {
        double r30007 = 3.0;
        double r30008 = 2.0;
        double r30009 = r;
        double r30010 = r30009 * r30009;
        double r30011 = r30008 / r30010;
        double r30012 = r30007 + r30011;
        double r30013 = 0.125;
        double r30014 = v;
        double r30015 = r30008 * r30014;
        double r30016 = r30007 - r30015;
        double r30017 = r30013 * r30016;
        double r30018 = w;
        double r30019 = r30018 * r30018;
        double r30020 = r30019 * r30009;
        double r30021 = r30020 * r30009;
        double r30022 = r30017 * r30021;
        double r30023 = 1.0;
        double r30024 = r30023 - r30014;
        double r30025 = r30022 / r30024;
        double r30026 = r30012 - r30025;
        double r30027 = 4.5;
        double r30028 = r30026 - r30027;
        return r30028;
}
↓
double f(double v, double w, double r) {
        double r30029 = 3.0;
        double r30030 = 2.0;
        double r30031 = r;
        double r30032 = r30031 * r30031;
        double r30033 = r30030 / r30032;
        double r30034 = 0.125;
        double r30035 = v;
        double r30036 = r30030 * r30035;
        double r30037 = r30029 - r30036;
        double r30038 = r30034 * r30037;
        double r30039 = 1.0;
        double r30040 = r30039 - r30035;
        double r30041 = r30038 / r30040;
        double r30042 = w;
        double r30043 = r30031 * r30042;
        double r30044 = fabs(r30043);
        double r30045 = 2.0;
        double r30046 = pow(r30044, r30045);
        double r30047 = r30041 * r30046;
        double r30048 = sqrt(r30047);
        double r30049 = r30048 * r30048;
        double r30050 = r30033 - r30049;
        double r30051 = r30029 + r30050;
        double r30052 = 4.5;
        double r30053 = r30051 - r30052;
        return r30053;
}

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. Initial program 12.8

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

  2. Simplified8.4

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

  4. Applied add-sqr-sqrt8.5

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

  5. Simplified8.4

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

  6. Simplified0.4

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

  8. Applied fma-udef0.4

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

  9. Applied associate--r+0.4

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

  10. Simplified0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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))