Average Error: 0.2 → 0.2
Time: 34.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(1 - a \cdot 3\right) \cdot \left(b \cdot b\right) + \left(a + 1\right) \cdot \left(a \cdot a\right)\right) \cdot 4 + \mathsf{fma}\left(b \cdot b, b \cdot b, \left(\left(a \cdot a + \left(b \cdot b\right) \cdot 2\right) \cdot a\right) \cdot a\right)\right) - 1\]
\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
↓
\left(\left(\left(1 - a \cdot 3\right) \cdot \left(b \cdot b\right) + \left(a + 1\right) \cdot \left(a \cdot a\right)\right) \cdot 4 + \mathsf{fma}\left(b \cdot b, b \cdot b, \left(\left(a \cdot a + \left(b \cdot b\right) \cdot 2\right) \cdot a\right) \cdot a\right)\right) - 1
double f(double a, double b) {
        double r5467653 = a;
        double r5467654 = r5467653 * r5467653;
        double r5467655 = b;
        double r5467656 = r5467655 * r5467655;
        double r5467657 = r5467654 + r5467656;
        double r5467658 = 2.0;
        double r5467659 = pow(r5467657, r5467658);
        double r5467660 = 4.0;
        double r5467661 = 1.0;
        double r5467662 = r5467661 + r5467653;
        double r5467663 = r5467654 * r5467662;
        double r5467664 = 3.0;
        double r5467665 = r5467664 * r5467653;
        double r5467666 = r5467661 - r5467665;
        double r5467667 = r5467656 * r5467666;
        double r5467668 = r5467663 + r5467667;
        double r5467669 = r5467660 * r5467668;
        double r5467670 = r5467659 + r5467669;
        double r5467671 = r5467670 - r5467661;
        return r5467671;
}
↓
double f(double a, double b) {
        double r5467672 = 1.0;
        double r5467673 = a;
        double r5467674 = 3.0;
        double r5467675 = r5467673 * r5467674;
        double r5467676 = r5467672 - r5467675;
        double r5467677 = b;
        double r5467678 = r5467677 * r5467677;
        double r5467679 = r5467676 * r5467678;
        double r5467680 = r5467673 + r5467672;
        double r5467681 = r5467673 * r5467673;
        double r5467682 = r5467680 * r5467681;
        double r5467683 = r5467679 + r5467682;
        double r5467684 = 4.0;
        double r5467685 = r5467683 * r5467684;
        double r5467686 = 2.0;
        double r5467687 = r5467678 * r5467686;
        double r5467688 = r5467681 + r5467687;
        double r5467689 = r5467688 * r5467673;
        double r5467690 = r5467689 * r5467673;
        double r5467691 = fma(r5467678, r5467678, r5467690);
        double r5467692 = r5467685 + r5467691;
        double r5467693 = r5467692 - r5467672;
        return r5467693;
}

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. Taylor expanded around -inf 0.0

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

  3. Simplified0.2

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

  5. Applied associate-*l*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))