Average Error: 0.2 → 0.7
Time: 6.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}}\right)}^{2} - 1\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}}\right)}^{2} - 1\right)
double f(double a, double b) {
        double r173960 = a;
        double r173961 = r173960 * r173960;
        double r173962 = b;
        double r173963 = r173962 * r173962;
        double r173964 = r173961 + r173963;
        double r173965 = 2.0;
        double r173966 = pow(r173964, r173965);
        double r173967 = 4.0;
        double r173968 = 1.0;
        double r173969 = r173968 + r173960;
        double r173970 = r173961 * r173969;
        double r173971 = 3.0;
        double r173972 = r173971 * r173960;
        double r173973 = r173968 - r173972;
        double r173974 = r173963 * r173973;
        double r173975 = r173970 + r173974;
        double r173976 = r173967 * r173975;
        double r173977 = r173966 + r173976;
        double r173978 = r173977 - r173968;
        return r173978;
}
↓
double f(double a, double b) {
        double r173979 = 4.0;
        double r173980 = a;
        double r173981 = r173980 * r173980;
        double r173982 = 1.0;
        double r173983 = r173982 + r173980;
        double r173984 = b;
        double r173985 = r173984 * r173984;
        double r173986 = 3.0;
        double r173987 = r173986 * r173980;
        double r173988 = r173982 - r173987;
        double r173989 = r173985 * r173988;
        double r173990 = fma(r173981, r173983, r173989);
        double r173991 = r173981 + r173985;
        double r173992 = cbrt(r173991);
        double r173993 = r173992 * r173992;
        double r173994 = 2.0;
        double r173995 = pow(r173993, r173994);
        double r173996 = cbrt(r173993);
        double r173997 = sqrt(r173991);
        double r173998 = cbrt(r173997);
        double r173999 = r173998 * r173998;
        double r174000 = cbrt(r173999);
        double r174001 = r173996 * r174000;
        double r174002 = pow(r174001, r173994);
        double r174003 = r173995 * r174002;
        double r174004 = r174003 - r173982;
        double r174005 = fma(r173979, r173990, r174004);
        return r174005;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\color{blue}{\left(\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right) \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}}^{2} - 1\right)\]

  5. Applied unpow-prod-down0.7

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

  7. Applied add-cube-cbrt0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right) \cdot \sqrt[3]{a \cdot a + b \cdot b}}}\right)}^{2} - 1\right)\]

  8. Applied cbrt-prod0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b}}\right)}}^{2} - 1\right)\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt[3]{\color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}}}}\right)}^{2} - 1\right)\]

  11. Applied cbrt-prod0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\color{blue}{\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}}}\right)}^{2} - 1\right)\]

  12. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}}\right)}^{2} - 1\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))