Average Error: 0.2 → 0.0
Time: 26.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(a \cdot 4\right) \cdot \left(b \cdot b + \left(a - a \cdot a\right)\right) + \left(-1 + \left(\left(b \cdot b\right) \cdot 12 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right)\]
\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
↓
\left(a \cdot 4\right) \cdot \left(b \cdot b + \left(a - a \cdot a\right)\right) + \left(-1 + \left(\left(b \cdot b\right) \cdot 12 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right)
double f(double a, double b) {
        double r9592406 = a;
        double r9592407 = r9592406 * r9592406;
        double r9592408 = b;
        double r9592409 = r9592408 * r9592408;
        double r9592410 = r9592407 + r9592409;
        double r9592411 = 2.0;
        double r9592412 = pow(r9592410, r9592411);
        double r9592413 = 4.0;
        double r9592414 = 1.0;
        double r9592415 = r9592414 - r9592406;
        double r9592416 = r9592407 * r9592415;
        double r9592417 = 3.0;
        double r9592418 = r9592417 + r9592406;
        double r9592419 = r9592409 * r9592418;
        double r9592420 = r9592416 + r9592419;
        double r9592421 = r9592413 * r9592420;
        double r9592422 = r9592412 + r9592421;
        double r9592423 = r9592422 - r9592414;
        return r9592423;
}
↓
double f(double a, double b) {
        double r9592424 = a;
        double r9592425 = 4.0;
        double r9592426 = r9592424 * r9592425;
        double r9592427 = b;
        double r9592428 = r9592427 * r9592427;
        double r9592429 = r9592424 * r9592424;
        double r9592430 = r9592424 - r9592429;
        double r9592431 = r9592428 + r9592430;
        double r9592432 = r9592426 * r9592431;
        double r9592433 = -1.0;
        double r9592434 = 12.0;
        double r9592435 = r9592428 * r9592434;
        double r9592436 = r9592429 + r9592428;
        double r9592437 = sqrt(r9592436);
        double r9592438 = pow(r9592437, r9592425);
        double r9592439 = r9592435 + r9592438;
        double r9592440 = r9592433 + r9592439;
        double r9592441 = r9592432 + r9592440;
        return r9592441;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied pow10.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied pow30.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))