Average Error: 0.2 → 0.2
Time: 20.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(1 + a\right) \cdot {a}^{2} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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(\left(1 + a\right) \cdot {a}^{2} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r108493 = a;
        double r108494 = r108493 * r108493;
        double r108495 = b;
        double r108496 = r108495 * r108495;
        double r108497 = r108494 + r108496;
        double r108498 = 2.0;
        double r108499 = pow(r108497, r108498);
        double r108500 = 4.0;
        double r108501 = 1.0;
        double r108502 = r108501 + r108493;
        double r108503 = r108494 * r108502;
        double r108504 = 3.0;
        double r108505 = r108504 * r108493;
        double r108506 = r108501 - r108505;
        double r108507 = r108496 * r108506;
        double r108508 = r108503 + r108507;
        double r108509 = r108500 * r108508;
        double r108510 = r108499 + r108509;
        double r108511 = r108510 - r108501;
        return r108511;
}
↓
double f(double a, double b) {
        double r108512 = a;
        double r108513 = r108512 * r108512;
        double r108514 = b;
        double r108515 = r108514 * r108514;
        double r108516 = r108513 + r108515;
        double r108517 = 2.0;
        double r108518 = pow(r108516, r108517);
        double r108519 = 4.0;
        double r108520 = 1.0;
        double r108521 = r108520 + r108512;
        double r108522 = 2.0;
        double r108523 = pow(r108512, r108522);
        double r108524 = r108521 * r108523;
        double r108525 = 3.0;
        double r108526 = r108525 * r108512;
        double r108527 = r108520 - r108526;
        double r108528 = r108515 * r108527;
        double r108529 = r108524 + r108528;
        double r108530 = r108519 * r108529;
        double r108531 = r108518 + r108530;
        double r108532 = r108531 - r108520;
        return r108532;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 
(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))