Average Error: 0.2 → 0.1
Time: 25.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\sqrt{b \cdot b + a \cdot a} \cdot \left(\sqrt{b \cdot b + a \cdot a} \cdot \left(b \cdot b + a \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot -12\right) + \left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4\right) - 1\]
\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(\sqrt{b \cdot b + a \cdot a} \cdot \left(\sqrt{b \cdot b + a \cdot a} \cdot \left(b \cdot b + a \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot -12\right) + \left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r11041685 = a;
        double r11041686 = r11041685 * r11041685;
        double r11041687 = b;
        double r11041688 = r11041687 * r11041687;
        double r11041689 = r11041686 + r11041688;
        double r11041690 = 2.0;
        double r11041691 = pow(r11041689, r11041690);
        double r11041692 = 4.0;
        double r11041693 = 1.0;
        double r11041694 = r11041693 + r11041685;
        double r11041695 = r11041686 * r11041694;
        double r11041696 = 3.0;
        double r11041697 = r11041696 * r11041685;
        double r11041698 = r11041693 - r11041697;
        double r11041699 = r11041688 * r11041698;
        double r11041700 = r11041695 + r11041699;
        double r11041701 = r11041692 * r11041700;
        double r11041702 = r11041691 + r11041701;
        double r11041703 = r11041702 - r11041693;
        return r11041703;
}
↓
double f(double a, double b) {
        double r11041704 = b;
        double r11041705 = r11041704 * r11041704;
        double r11041706 = a;
        double r11041707 = r11041706 * r11041706;
        double r11041708 = r11041705 + r11041707;
        double r11041709 = sqrt(r11041708);
        double r11041710 = r11041709 * r11041708;
        double r11041711 = r11041709 * r11041710;
        double r11041712 = r11041705 * r11041706;
        double r11041713 = -12.0;
        double r11041714 = r11041712 * r11041713;
        double r11041715 = r11041711 + r11041714;
        double r11041716 = r11041707 * r11041706;
        double r11041717 = r11041716 + r11041708;
        double r11041718 = 4.0;
        double r11041719 = r11041717 * r11041718;
        double r11041720 = r11041715 + r11041719;
        double r11041721 = 1.0;
        double r11041722 = r11041720 - r11041721;
        return r11041722;
}

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

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  5. Applied associate-*r*0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  6. Final simplification0.1

    \[\leadsto \left(\left(\sqrt{b \cdot b + a \cdot a} \cdot \left(\sqrt{b \cdot b + a \cdot a} \cdot \left(b \cdot b + a \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot -12\right) + \left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4\right) - 1\]

Reproduce

herbie shell --seed 2019149 
(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))