Rosa's FloatVsDoubleBenchmark

Average Error: 0.5 → 0.7

Time: 1.0m

Precision: 64

Math

\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

↓

\[\left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1} \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}}{\sqrt{1 + x1 \cdot x1}}, \frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}}{\sqrt{1 + x1 \cdot x1}}, -3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}, \sqrt{6} \cdot \left(-\sqrt{6}\right)\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right)\right)\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3\right) + x1\]

C

double f(double x1, double x2) {
        double r3290900 = x1;
        double r3290901 = 2.0;
        double r3290902 = r3290901 * r3290900;
        double r3290903 = 3.0;
        double r3290904 = r3290903 * r3290900;
        double r3290905 = r3290904 * r3290900;
        double r3290906 = x2;
        double r3290907 = r3290901 * r3290906;
        double r3290908 = r3290905 + r3290907;
        double r3290909 = r3290908 - r3290900;
        double r3290910 = r3290900 * r3290900;
        double r3290911 = 1.0;
        double r3290912 = r3290910 + r3290911;
        double r3290913 = r3290909 / r3290912;
        double r3290914 = r3290902 * r3290913;
        double r3290915 = r3290913 - r3290903;
        double r3290916 = r3290914 * r3290915;
        double r3290917 = 4.0;
        double r3290918 = r3290917 * r3290913;
        double r3290919 = 6.0;
        double r3290920 = r3290918 - r3290919;
        double r3290921 = r3290910 * r3290920;
        double r3290922 = r3290916 + r3290921;
        double r3290923 = r3290922 * r3290912;
        double r3290924 = r3290905 * r3290913;
        double r3290925 = r3290923 + r3290924;
        double r3290926 = r3290910 * r3290900;
        double r3290927 = r3290925 + r3290926;
        double r3290928 = r3290927 + r3290900;
        double r3290929 = r3290905 - r3290907;
        double r3290930 = r3290929 - r3290900;
        double r3290931 = r3290930 / r3290912;
        double r3290932 = r3290903 * r3290931;
        double r3290933 = r3290928 + r3290932;
        double r3290934 = r3290900 + r3290933;
        return r3290934;
}

↓

double f(double x1, double x2) {
        double r3290935 = x1;
        double r3290936 = r3290935 * r3290935;
        double r3290937 = r3290935 * r3290936;
        double r3290938 = 2.0;
        double r3290939 = r3290935 * r3290938;
        double r3290940 = 3.0;
        double r3290941 = r3290940 * r3290935;
        double r3290942 = r3290941 * r3290935;
        double r3290943 = x2;
        double r3290944 = r3290943 * r3290938;
        double r3290945 = r3290942 + r3290944;
        double r3290946 = r3290945 - r3290935;
        double r3290947 = 1.0;
        double r3290948 = r3290947 + r3290936;
        double r3290949 = r3290946 / r3290948;
        double r3290950 = r3290939 * r3290949;
        double r3290951 = cbrt(r3290946);
        double r3290952 = r3290951 * r3290951;
        double r3290953 = sqrt(r3290948);
        double r3290954 = r3290952 / r3290953;
        double r3290955 = r3290951 / r3290953;
        double r3290956 = -r3290940;
        double r3290957 = fma(r3290954, r3290955, r3290956);
        double r3290958 = r3290950 * r3290957;
        double r3290959 = 6.0;
        double r3290960 = -1.0;
        double r3290961 = fma(r3290959, r3290960, r3290959);
        double r3290962 = r3290936 * r3290961;
        double r3290963 = 4.0;
        double r3290964 = sqrt(r3290959);
        double r3290965 = -r3290964;
        double r3290966 = r3290964 * r3290965;
        double r3290967 = fma(r3290963, r3290949, r3290966);
        double r3290968 = r3290936 * r3290967;
        double r3290969 = r3290962 + r3290968;
        double r3290970 = r3290958 + r3290969;
        double r3290971 = r3290970 * r3290948;
        double r3290972 = r3290942 * r3290949;
        double r3290973 = r3290971 + r3290972;
        double r3290974 = r3290937 + r3290973;
        double r3290975 = r3290935 + r3290974;
        double r3290976 = r3290942 - r3290944;
        double r3290977 = r3290976 - r3290935;
        double r3290978 = r3290977 / r3290948;
        double r3290979 = r3290978 * r3290940;
        double r3290980 = r3290975 + r3290979;
        double r3290981 = r3290980 + r3290935;
        return r3290981;
}

Error

Bits error versus x1

Bits error versus x2

Derivation

1. Initial program 0.5

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.5

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - \color{blue}{\sqrt{6} \cdot \sqrt{6}}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

4. Applied prod-diff 0.5

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \mathsf{fma}\left(-\sqrt{6}, \sqrt{6}, \sqrt{6} \cdot \sqrt{6}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

5. Applied distribute-lft-in 0.5

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(-\sqrt{6}, \sqrt{6}, \sqrt{6} \cdot \sqrt{6}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

6. Simplified 0.5

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \color{blue}{\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right)}\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.6

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{\color{blue}{\sqrt{x1 \cdot x1 + 1} \cdot \sqrt{x1 \cdot x1 + 1}}} - 3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

9. Applied add-cube-cbrt 0.7

   \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1} \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}\right) \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}}}{\sqrt{x1 \cdot x1 + 1} \cdot \sqrt{x1 \cdot x1 + 1}} - 3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

10. Applied times-frac 0.7

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\color{blue}{\frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1} \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}}{\sqrt{x1 \cdot x1 + 1}} \cdot \frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}}{\sqrt{x1 \cdot x1 + 1}}} - 3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

11. Applied fma-neg 0.7

    \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1} \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}}{\sqrt{x1 \cdot x1 + 1}}, \frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}}{\sqrt{x1 \cdot x1 + 1}}, -3\right)} + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}, -\sqrt{6} \cdot \sqrt{6}\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right)\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

12. Final simplification 0.7

    \[\leadsto \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1} \cdot \sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}}{\sqrt{1 + x1 \cdot x1}}, \frac{\sqrt[3]{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}}{\sqrt{1 + x1 \cdot x1}}, -3\right) + \left(\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(6, -1, 6\right) + \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(4, \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}, \sqrt{6} \cdot \left(-\sqrt{6}\right)\right)\right)\right) \cdot \left(1 + x1 \cdot x1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + x2 \cdot 2\right) - x1}{1 + x1 \cdot x1}\right)\right)\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{1 + x1 \cdot x1} \cdot 3\right) + x1\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))