Average Error: 0.2 → 0.2
Time: 6.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\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(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt{{\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
double f(double a, double b) {
        double r258669 = a;
        double r258670 = r258669 * r258669;
        double r258671 = b;
        double r258672 = r258671 * r258671;
        double r258673 = r258670 + r258672;
        double r258674 = 2.0;
        double r258675 = pow(r258673, r258674);
        double r258676 = 4.0;
        double r258677 = 1.0;
        double r258678 = r258677 - r258669;
        double r258679 = r258670 * r258678;
        double r258680 = 3.0;
        double r258681 = r258680 + r258669;
        double r258682 = r258672 * r258681;
        double r258683 = r258679 + r258682;
        double r258684 = r258676 * r258683;
        double r258685 = r258675 + r258684;
        double r258686 = r258685 - r258677;
        return r258686;
}
↓
double f(double a, double b) {
        double r258687 = a;
        double r258688 = r258687 * r258687;
        double r258689 = b;
        double r258690 = r258689 * r258689;
        double r258691 = r258688 + r258690;
        double r258692 = 2.0;
        double r258693 = pow(r258691, r258692);
        double r258694 = sqrt(r258693);
        double r258695 = 4.0;
        double r258696 = 1.0;
        double r258697 = r258696 - r258687;
        double r258698 = r258688 * r258697;
        double r258699 = 3.0;
        double r258700 = r258699 + r258687;
        double r258701 = r258690 * r258700;
        double r258702 = r258698 + r258701;
        double r258703 = r258695 * r258702;
        double r258704 = fma(r258694, r258694, r258703);
        double r258705 = r258704 - r258696;
        return r258705;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +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))