Average Error: 0.2 → 0.0
Time: 16.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[(\left(b \cdot b\right) \cdot 4 + -1)_* + {\left(\sqrt{a^2 + b^2}^*\right)}^{4}\]
double f(double a, double b) {
        double r33692771 = a;
        double r33692772 = r33692771 * r33692771;
        double r33692773 = b;
        double r33692774 = r33692773 * r33692773;
        double r33692775 = r33692772 + r33692774;
        double r33692776 = 2.0;
        double r33692777 = pow(r33692775, r33692776);
        double r33692778 = 4.0;
        double r33692779 = r33692778 * r33692774;
        double r33692780 = r33692777 + r33692779;
        double r33692781 = 1.0;
        double r33692782 = r33692780 - r33692781;
        return r33692782;
}
↓
double f(double a, double b) {
        double r33692783 = b;
        double r33692784 = r33692783 * r33692783;
        double r33692785 = 4.0;
        double r33692786 = -1.0;
        double r33692787 = fma(r33692784, r33692785, r33692786);
        double r33692788 = a;
        double r33692789 = hypot(r33692788, r33692783);
        double r33692790 = pow(r33692789, r33692785);
        double r33692791 = r33692787 + r33692790;
        return r33692791;
}


\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
(\left(b \cdot b\right) \cdot 4 + -1)_* + {\left(\sqrt{a^2 + b^2}^*\right)}^{4}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{(\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*}\]
  3. Using strategy rm

  4. Applied fma-udef0.2

    \[\leadsto \color{blue}{(a \cdot a + \left(b \cdot b\right))_* \cdot (a \cdot a + \left(b \cdot b\right))_* + (\left(b \cdot b\right) \cdot 4 + -1)_*}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.2

    \[\leadsto \color{blue}{\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)} \cdot (a \cdot a + \left(b \cdot b\right))_* + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  7. Applied associate-*l*0.1

    \[\leadsto \color{blue}{\sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot (a \cdot a + \left(b \cdot b\right))_*\right)} + (\left(b \cdot b\right) \cdot 4 + -1)_*\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.1

    \[\leadsto \sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \color{blue}{\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)}\right) + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  10. Applied cube-unmult0.1

    \[\leadsto \sqrt{(a \cdot a + \left(b \cdot b\right))_*} \cdot \color{blue}{{\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)}^{3}} + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  11. Applied pow10.1

    \[\leadsto \color{blue}{{\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)}^{1}} \cdot {\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)}^{3} + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  12. Applied pow-prod-up0.0

    \[\leadsto \color{blue}{{\left(\sqrt{(a \cdot a + \left(b \cdot b\right))_*}\right)}^{\left(1 + 3\right)}} + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  13. Simplified0.0

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

  14. Simplified0.0

    \[\leadsto {\left(\sqrt{a^2 + b^2}^*\right)}^{\color{blue}{4}} + (\left(b \cdot b\right) \cdot 4 + -1)_*\]

  15. Final simplification0.0

    \[\leadsto (\left(b \cdot b\right) \cdot 4 + -1)_* + {\left(\sqrt{a^2 + b^2}^*\right)}^{4}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))