Average Error: 0.2 → 0.2
Time: 26.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)}, 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right)}, 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r219662 = a;
        double r219663 = r219662 * r219662;
        double r219664 = b;
        double r219665 = r219664 * r219664;
        double r219666 = r219663 + r219665;
        double r219667 = 2.0;
        double r219668 = pow(r219666, r219667);
        double r219669 = 4.0;
        double r219670 = 1.0;
        double r219671 = r219670 - r219662;
        double r219672 = r219663 * r219671;
        double r219673 = 3.0;
        double r219674 = r219673 + r219662;
        double r219675 = r219665 * r219674;
        double r219676 = r219672 + r219675;
        double r219677 = r219669 * r219676;
        double r219678 = r219668 + r219677;
        double r219679 = r219678 - r219670;
        return r219679;
}
↓
double f(double a, double b) {
        double r219680 = a;
        double r219681 = r219680 * r219680;
        double r219682 = 1.0;
        double r219683 = r219682 - r219680;
        double r219684 = b;
        double r219685 = r219684 * r219684;
        double r219686 = 3.0;
        double r219687 = r219686 + r219680;
        double r219688 = r219685 * r219687;
        double r219689 = fma(r219681, r219683, r219688);
        double r219690 = cbrt(r219689);
        double r219691 = r219690 * r219690;
        double r219692 = r219691 * r219690;
        double r219693 = 4.0;
        double r219694 = fma(r219680, r219680, r219685);
        double r219695 = 2.0;
        double r219696 = pow(r219694, r219695);
        double r219697 = fma(r219692, r219693, r219696);
        double r219698 = r219697 - r219682;
        return r219698;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))