Average Error: 12.5 → 0.5
Time: 1.3m
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\sqrt{\frac{\frac{\sqrt{2}}{r}}{r}} \cdot \left(\sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\sqrt{2}}\right) + 3\right) - (\left(\frac{(v \cdot -2 + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]
double f(double v, double w, double r) {
        double r3572001 = 3.0;
        double r3572002 = 2.0;
        double r3572003 = r;
        double r3572004 = r3572003 * r3572003;
        double r3572005 = r3572002 / r3572004;
        double r3572006 = r3572001 + r3572005;
        double r3572007 = 0.125;
        double r3572008 = v;
        double r3572009 = r3572002 * r3572008;
        double r3572010 = r3572001 - r3572009;
        double r3572011 = r3572007 * r3572010;
        double r3572012 = w;
        double r3572013 = r3572012 * r3572012;
        double r3572014 = r3572013 * r3572003;
        double r3572015 = r3572014 * r3572003;
        double r3572016 = r3572011 * r3572015;
        double r3572017 = 1.0;
        double r3572018 = r3572017 - r3572008;
        double r3572019 = r3572016 / r3572018;
        double r3572020 = r3572006 - r3572019;
        double r3572021 = 4.5;
        double r3572022 = r3572020 - r3572021;
        return r3572022;
}
↓
double f(double v, double w, double r) {
        double r3572023 = 2.0;
        double r3572024 = sqrt(r3572023);
        double r3572025 = r;
        double r3572026 = r3572024 / r3572025;
        double r3572027 = r3572026 / r3572025;
        double r3572028 = sqrt(r3572027);
        double r3572029 = r3572023 / r3572025;
        double r3572030 = r3572029 / r3572025;
        double r3572031 = sqrt(r3572030);
        double r3572032 = sqrt(r3572024);
        double r3572033 = r3572031 * r3572032;
        double r3572034 = r3572028 * r3572033;
        double r3572035 = 3.0;
        double r3572036 = r3572034 + r3572035;
        double r3572037 = v;
        double r3572038 = -2.0;
        double r3572039 = fma(r3572037, r3572038, r3572035);
        double r3572040 = 1.0;
        double r3572041 = r3572040 - r3572037;
        double r3572042 = 0.125;
        double r3572043 = r3572041 / r3572042;
        double r3572044 = r3572039 / r3572043;
        double r3572045 = w;
        double r3572046 = r3572025 * r3572045;
        double r3572047 = r3572046 * r3572046;
        double r3572048 = 4.5;
        double r3572049 = fma(r3572044, r3572047, r3572048);
        double r3572050 = r3572036 - r3572049;
        return r3572050;
}


\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(\sqrt{\frac{\frac{\sqrt{2}}{r}}{r}} \cdot \left(\sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\sqrt{2}}\right) + 3\right) - (\left(\frac{(v \cdot -2 + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  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. Simplified0.4

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

  4. Applied associate-/r*0.4

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

  6. Applied add-sqr-sqrt0.5

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

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

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

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

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

  10. Applied add-sqr-sqrt0.6

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

  11. Applied times-frac0.5

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

  12. Applied times-frac0.5

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

  13. Applied sqrt-prod0.5

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

  14. Applied associate-*r*0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(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))