Average Error: 0.2 → 0.2
Time: 12.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(\sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)} \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(\sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)} \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) \cdot \sqrt[3]{\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 r313626 = a;
        double r313627 = r313626 * r313626;
        double r313628 = b;
        double r313629 = r313628 * r313628;
        double r313630 = r313627 + r313629;
        double r313631 = 2.0;
        double r313632 = pow(r313630, r313631);
        double r313633 = 4.0;
        double r313634 = 1.0;
        double r313635 = r313634 + r313626;
        double r313636 = r313627 * r313635;
        double r313637 = 3.0;
        double r313638 = r313637 * r313626;
        double r313639 = r313634 - r313638;
        double r313640 = r313629 * r313639;
        double r313641 = r313636 + r313640;
        double r313642 = r313633 * r313641;
        double r313643 = r313632 + r313642;
        double r313644 = r313643 - r313634;
        return r313644;
}
↓
double f(double a, double b) {
        double r313645 = 4.0;
        double r313646 = a;
        double r313647 = r313646 * r313646;
        double r313648 = 1.0;
        double r313649 = r313648 + r313646;
        double r313650 = b;
        double r313651 = r313650 * r313650;
        double r313652 = 3.0;
        double r313653 = r313652 * r313646;
        double r313654 = r313648 - r313653;
        double r313655 = r313651 * r313654;
        double r313656 = cbrt(r313655);
        double r313657 = r313656 * r313656;
        double r313658 = r313657 * r313656;
        double r313659 = fma(r313647, r313649, r313658);
        double r313660 = fma(r313646, r313646, r313651);
        double r313661 = 2.0;
        double r313662 = pow(r313660, r313661);
        double r313663 = fma(r313645, r313659, r313662);
        double r313664 = r313663 - r313648;
        return r313664;
}

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-cube-cbrt0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +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))