Average Error: 0.2 → 0.0
Time: 18.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - \left(1 - \left(\left(\left(a + 3\right) \cdot \left(b \cdot b\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4\right)\]
\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(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - \left(1 - \left(\left(\left(a + 3\right) \cdot \left(b \cdot b\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4\right)
double f(double a, double b) {
        double r10362339 = a;
        double r10362340 = r10362339 * r10362339;
        double r10362341 = b;
        double r10362342 = r10362341 * r10362341;
        double r10362343 = r10362340 + r10362342;
        double r10362344 = 2.0;
        double r10362345 = pow(r10362343, r10362344);
        double r10362346 = 4.0;
        double r10362347 = 1.0;
        double r10362348 = r10362347 - r10362339;
        double r10362349 = r10362340 * r10362348;
        double r10362350 = 3.0;
        double r10362351 = r10362350 + r10362339;
        double r10362352 = r10362342 * r10362351;
        double r10362353 = r10362349 + r10362352;
        double r10362354 = r10362346 * r10362353;
        double r10362355 = r10362345 + r10362354;
        double r10362356 = r10362355 - r10362347;
        return r10362356;
}
↓
double f(double a, double b) {
        double r10362357 = a;
        double r10362358 = r10362357 * r10362357;
        double r10362359 = b;
        double r10362360 = r10362359 * r10362359;
        double r10362361 = r10362358 + r10362360;
        double r10362362 = sqrt(r10362361);
        double r10362363 = 4.0;
        double r10362364 = pow(r10362362, r10362363);
        double r10362365 = 1.0;
        double r10362366 = 3.0;
        double r10362367 = r10362357 + r10362366;
        double r10362368 = r10362367 * r10362360;
        double r10362369 = r10362368 + r10362358;
        double r10362370 = r10362357 * r10362358;
        double r10362371 = r10362369 - r10362370;
        double r10362372 = r10362371 * r10362363;
        double r10362373 = r10362365 - r10362372;
        double r10362374 = r10362364 - r10362373;
        return r10362374;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 
(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))