Rosa's TurbineBenchmark

Average Error: 13.0 → 0.4

Time: 27.4s

Precision: 64

Math

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

↓

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

C

double f(double v, double w, double r) {
        double r40781 = 3.0;
        double r40782 = 2.0;
        double r40783 = r;
        double r40784 = r40783 * r40783;
        double r40785 = r40782 / r40784;
        double r40786 = r40781 + r40785;
        double r40787 = 0.125;
        double r40788 = v;
        double r40789 = r40782 * r40788;
        double r40790 = r40781 - r40789;
        double r40791 = r40787 * r40790;
        double r40792 = w;
        double r40793 = r40792 * r40792;
        double r40794 = r40793 * r40783;
        double r40795 = r40794 * r40783;
        double r40796 = r40791 * r40795;
        double r40797 = 1.0;
        double r40798 = r40797 - r40788;
        double r40799 = r40796 / r40798;
        double r40800 = r40786 - r40799;
        double r40801 = 4.5;
        double r40802 = r40800 - r40801;
        return r40802;
}

↓

double f(double v, double w, double r) {
        double r40803 = 3.0;
        double r40804 = 2.0;
        double r40805 = r;
        double r40806 = r40805 * r40805;
        double r40807 = r40804 / r40806;
        double r40808 = r40803 + r40807;
        double r40809 = w;
        double r40810 = r40809 * r40805;
        double r40811 = fabs(r40810);
        double r40812 = fabs(r40811);
        double r40813 = 0.125;
        double r40814 = r40812 * r40813;
        double r40815 = sqrt(r40814);
        double r40816 = v;
        double r40817 = r40804 * r40816;
        double r40818 = r40803 - r40817;
        double r40819 = 1.0;
        double r40820 = r40819 - r40816;
        double r40821 = r40818 / r40820;
        double r40822 = r40812 * r40821;
        double r40823 = sqrt(r40822);
        double r40824 = r40815 * r40823;
        double r40825 = r40813 * r40811;
        double r40826 = r40820 / r40811;
        double r40827 = r40818 / r40826;
        double r40828 = r40825 * r40827;
        double r40829 = sqrt(r40828);
        double r40830 = r40824 * r40829;
        double r40831 = r40808 - r40830;
        double r40832 = 4.5;
        double r40833 = r40831 - r40832;
        return r40833;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

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 add-sqr-sqrt 13.0

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

4. Simplified 13.0

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

5. Simplified 7.0

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

7. Applied add-sqr-sqrt 7.0

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

8. Simplified 7.0

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

9. Simplified 0.4

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied unpow-prod-down 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied times-frac 0.5

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

15. Applied times-frac 0.5

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

16. Simplified 0.4

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

17. Simplified 0.4

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

19. Applied add-sqr-sqrt 0.4

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

20. Applied *-un-lft-identity 0.4

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

21. Applied times-frac 0.4

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

22. Applied times-frac 0.4

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

23. Applied sqrt-prod 0.5

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

24. Simplified 0.5

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

25. Simplified 0.4

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

26. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019325 
(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))