Average Error: 0.2 → 0.2
Time: 29.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\]
\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
↓
\sqrt{\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(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} \cdot \sqrt{\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(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1
double f(double a, double b) {
        double r302800 = a;
        double r302801 = r302800 * r302800;
        double r302802 = b;
        double r302803 = r302802 * r302802;
        double r302804 = r302801 + r302803;
        double r302805 = 2.0;
        double r302806 = pow(r302804, r302805);
        double r302807 = 4.0;
        double r302808 = 1.0;
        double r302809 = r302808 + r302800;
        double r302810 = r302801 * r302809;
        double r302811 = 3.0;
        double r302812 = r302811 * r302800;
        double r302813 = r302808 - r302812;
        double r302814 = r302803 * r302813;
        double r302815 = r302810 + r302814;
        double r302816 = r302807 * r302815;
        double r302817 = r302806 + r302816;
        double r302818 = r302817 - r302808;
        return r302818;
}
↓
double f(double a, double b) {
        double r302819 = 4.0;
        double r302820 = a;
        double r302821 = r302820 * r302820;
        double r302822 = 1.0;
        double r302823 = r302822 + r302820;
        double r302824 = b;
        double r302825 = r302824 * r302824;
        double r302826 = 3.0;
        double r302827 = r302826 * r302820;
        double r302828 = r302822 - r302827;
        double r302829 = r302825 * r302828;
        double r302830 = fma(r302821, r302823, r302829);
        double r302831 = fma(r302820, r302820, r302825);
        double r302832 = 2.0;
        double r302833 = pow(r302831, r302832);
        double r302834 = fma(r302819, r302830, r302833);
        double r302835 = sqrt(r302834);
        double r302836 = r302835 * r302835;
        double r302837 = r302836 - r302822;
        return r302837;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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))