Average Error: 0.2 → 0.2
Time: 5.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r293773 = a;
        double r293774 = r293773 * r293773;
        double r293775 = b;
        double r293776 = r293775 * r293775;
        double r293777 = r293774 + r293776;
        double r293778 = 2.0;
        double r293779 = pow(r293777, r293778);
        double r293780 = 4.0;
        double r293781 = 1.0;
        double r293782 = r293781 + r293773;
        double r293783 = r293774 * r293782;
        double r293784 = 3.0;
        double r293785 = r293784 * r293773;
        double r293786 = r293781 - r293785;
        double r293787 = r293776 * r293786;
        double r293788 = r293783 + r293787;
        double r293789 = r293780 * r293788;
        double r293790 = r293779 + r293789;
        double r293791 = r293790 - r293781;
        return r293791;
}
↓
double f(double a, double b) {
        double r293792 = a;
        double r293793 = r293792 * r293792;
        double r293794 = b;
        double r293795 = r293794 * r293794;
        double r293796 = r293793 + r293795;
        double r293797 = 2.0;
        double r293798 = pow(r293796, r293797);
        double r293799 = 4.0;
        double r293800 = 1.0;
        double r293801 = r293800 * r293800;
        double r293802 = r293801 - r293793;
        double r293803 = r293793 * r293802;
        double r293804 = r293800 - r293792;
        double r293805 = r293803 / r293804;
        double r293806 = 3.0;
        double r293807 = r293806 * r293792;
        double r293808 = r293800 - r293807;
        double r293809 = r293795 * r293808;
        double r293810 = r293805 + r293809;
        double r293811 = r293799 * r293810;
        double r293812 = r293798 + r293811;
        double r293813 = r293812 - r293800;
        return r293813;
}

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. Using strategy rm

  3. Applied flip-+0.2

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

  4. Applied associate-*r/0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020039 
(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))