Average Error: 0.2 → 0.1
Time: 25.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1
double f(double a, double b) {
        double r9979418 = a;
        double r9979419 = r9979418 * r9979418;
        double r9979420 = b;
        double r9979421 = r9979420 * r9979420;
        double r9979422 = r9979419 + r9979421;
        double r9979423 = 2.0;
        double r9979424 = pow(r9979422, r9979423);
        double r9979425 = 4.0;
        double r9979426 = 1.0;
        double r9979427 = r9979426 + r9979418;
        double r9979428 = r9979419 * r9979427;
        double r9979429 = 3.0;
        double r9979430 = r9979429 * r9979418;
        double r9979431 = r9979426 - r9979430;
        double r9979432 = r9979421 * r9979431;
        double r9979433 = r9979428 + r9979432;
        double r9979434 = r9979425 * r9979433;
        double r9979435 = r9979424 + r9979434;
        double r9979436 = r9979435 - r9979426;
        return r9979436;
}
↓
double f(double a, double b) {
        double r9979437 = a;
        double r9979438 = r9979437 * r9979437;
        double r9979439 = r9979438 * r9979437;
        double r9979440 = b;
        double r9979441 = r9979440 * r9979440;
        double r9979442 = r9979441 + r9979438;
        double r9979443 = r9979439 + r9979442;
        double r9979444 = 4.0;
        double r9979445 = r9979443 * r9979444;
        double r9979446 = r9979441 * r9979437;
        double r9979447 = -12.0;
        double r9979448 = r9979446 * r9979447;
        double r9979449 = sqrt(r9979442);
        double r9979450 = 3.0;
        double r9979451 = pow(r9979449, r9979450);
        double r9979452 = r9979451 * r9979449;
        double r9979453 = r9979448 + r9979452;
        double r9979454 = r9979445 + r9979453;
        double r9979455 = 1.0;
        double r9979456 = r9979454 - r9979455;
        return r9979456;
}

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

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  9. Final simplification0.1

    \[\leadsto \left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019151 
(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))