Average Error: 0.2 → 0.0
Time: 25.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r9694395 = a;
        double r9694396 = r9694395 * r9694395;
        double r9694397 = b;
        double r9694398 = r9694397 * r9694397;
        double r9694399 = r9694396 + r9694398;
        double r9694400 = 2.0;
        double r9694401 = pow(r9694399, r9694400);
        double r9694402 = 4.0;
        double r9694403 = r9694402 * r9694398;
        double r9694404 = r9694401 + r9694403;
        double r9694405 = 1.0;
        double r9694406 = r9694404 - r9694405;
        return r9694406;
}
↓
double f(double a, double b) {
        double r9694407 = 4.0;
        double r9694408 = b;
        double r9694409 = r9694408 * r9694408;
        double r9694410 = r9694407 * r9694409;
        double r9694411 = a;
        double r9694412 = r9694411 * r9694411;
        double r9694413 = r9694409 + r9694412;
        double r9694414 = sqrt(r9694413);
        double r9694415 = pow(r9694414, r9694407);
        double r9694416 = r9694410 + r9694415;
        double r9694417 = 1.0;
        double r9694418 = r9694416 - r9694417;
        return r9694418;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))