Average Error: 0.2 → 0.2
Time: 7.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} - 1\]
\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
↓
\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)} \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)} - 1
double f(double a, double b) {
        double r322656 = a;
        double r322657 = r322656 * r322656;
        double r322658 = b;
        double r322659 = r322658 * r322658;
        double r322660 = r322657 + r322659;
        double r322661 = 2.0;
        double r322662 = pow(r322660, r322661);
        double r322663 = 4.0;
        double r322664 = 1.0;
        double r322665 = r322664 - r322656;
        double r322666 = r322657 * r322665;
        double r322667 = 3.0;
        double r322668 = r322667 + r322656;
        double r322669 = r322659 * r322668;
        double r322670 = r322666 + r322669;
        double r322671 = r322663 * r322670;
        double r322672 = r322662 + r322671;
        double r322673 = r322672 - r322664;
        return r322673;
}
↓
double f(double a, double b) {
        double r322674 = a;
        double r322675 = r322674 * r322674;
        double r322676 = b;
        double r322677 = r322676 * r322676;
        double r322678 = r322675 + r322677;
        double r322679 = 2.0;
        double r322680 = pow(r322678, r322679);
        double r322681 = 4.0;
        double r322682 = 1.0;
        double r322683 = r322682 - r322674;
        double r322684 = r322675 * r322683;
        double r322685 = 3.0;
        double r322686 = r322685 + r322674;
        double r322687 = r322677 * r322686;
        double r322688 = r322684 + r322687;
        double r322689 = r322681 * r322688;
        double r322690 = r322680 + r322689;
        double r322691 = sqrt(r322690);
        double r322692 = r322691 * r322691;
        double r322693 = r322692 - r322682;
        return r322693;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 
(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))