Rosa's TurbineBenchmark

Average Error: 13.0 → 0.4

Time: 27.3s

Precision: 64

Math

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

↓

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

C

double f(double v, double w, double r) {
        double r41049 = 3.0;
        double r41050 = 2.0;
        double r41051 = r;
        double r41052 = r41051 * r41051;
        double r41053 = r41050 / r41052;
        double r41054 = r41049 + r41053;
        double r41055 = 0.125;
        double r41056 = v;
        double r41057 = r41050 * r41056;
        double r41058 = r41049 - r41057;
        double r41059 = r41055 * r41058;
        double r41060 = w;
        double r41061 = r41060 * r41060;
        double r41062 = r41061 * r41051;
        double r41063 = r41062 * r41051;
        double r41064 = r41059 * r41063;
        double r41065 = 1.0;
        double r41066 = r41065 - r41056;
        double r41067 = r41064 / r41066;
        double r41068 = r41054 - r41067;
        double r41069 = 4.5;
        double r41070 = r41068 - r41069;
        return r41070;
}

↓

double f(double v, double w, double r) {
        double r41071 = 3.0;
        double r41072 = 2.0;
        double r41073 = r;
        double r41074 = r41073 * r41073;
        double r41075 = r41072 / r41074;
        double r41076 = r41071 + r41075;
        double r41077 = 0.125;
        double r41078 = w;
        double r41079 = r41078 * r41073;
        double r41080 = fabs(r41079);
        double r41081 = r41077 * r41080;
        double r41082 = v;
        double r41083 = r41072 * r41082;
        double r41084 = r41071 - r41083;
        double r41085 = 1.0;
        double r41086 = r41085 - r41082;
        double r41087 = r41086 / r41080;
        double r41088 = r41084 / r41087;
        double r41089 = r41081 * r41088;
        double r41090 = sqrt(r41089);
        double r41091 = fabs(r41080);
        double r41092 = r41091 * r41077;
        double r41093 = sqrt(r41092);
        double r41094 = r41084 / r41086;
        double r41095 = r41091 * r41094;
        double r41096 = sqrt(r41095);
        double r41097 = r41093 * r41096;
        double r41098 = r41090 * r41097;
        double r41099 = r41076 - r41098;
        double r41100 = 4.5;
        double r41101 = r41099 - r41100;
        return r41101;
}

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

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

12. Applied unpow-prod-down 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied times-frac 0.5

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

15. Applied times-frac 0.5

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

16. Simplified 0.4

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

17. Simplified 0.4

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

19. Applied add-sqr-sqrt 0.4

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

20. Applied *-un-lft-identity 0.4

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

21. Applied times-frac 0.4

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

22. Applied times-frac 0.4

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

23. Applied sqrt-prod 0.5

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

24. Simplified 0.5

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

25. Simplified 0.4

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

26. Final simplification 0.4

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

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))