Average Error: 0.2 → 0.2
Time: 23.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 + a\right)\right) + \left(\sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)} \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right)\right) - 1\]
\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(a \cdot \left(a \cdot \left(1 + a\right)\right) + \left(\sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)} \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right) \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)}\right)\right) - 1
double f(double a, double b) {
        double r204453 = a;
        double r204454 = r204453 * r204453;
        double r204455 = b;
        double r204456 = r204455 * r204455;
        double r204457 = r204454 + r204456;
        double r204458 = 2.0;
        double r204459 = pow(r204457, r204458);
        double r204460 = 4.0;
        double r204461 = 1.0;
        double r204462 = r204461 + r204453;
        double r204463 = r204454 * r204462;
        double r204464 = 3.0;
        double r204465 = r204464 * r204453;
        double r204466 = r204461 - r204465;
        double r204467 = r204456 * r204466;
        double r204468 = r204463 + r204467;
        double r204469 = r204460 * r204468;
        double r204470 = r204459 + r204469;
        double r204471 = r204470 - r204461;
        return r204471;
}
↓
double f(double a, double b) {
        double r204472 = a;
        double r204473 = r204472 * r204472;
        double r204474 = b;
        double r204475 = r204474 * r204474;
        double r204476 = r204473 + r204475;
        double r204477 = 2.0;
        double r204478 = pow(r204476, r204477);
        double r204479 = 4.0;
        double r204480 = 1.0;
        double r204481 = r204480 + r204472;
        double r204482 = r204472 * r204481;
        double r204483 = r204472 * r204482;
        double r204484 = 3.0;
        double r204485 = r204484 * r204472;
        double r204486 = r204480 - r204485;
        double r204487 = r204475 * r204486;
        double r204488 = cbrt(r204487);
        double r204489 = r204488 * r204488;
        double r204490 = r204489 * r204488;
        double r204491 = r204483 + r204490;
        double r204492 = r204479 * r204491;
        double r204493 = r204478 + r204492;
        double r204494 = r204493 - r204480;
        return r204494;
}

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. Using strategy rm

  3. Applied associate-*l*0.2

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

  5. Applied add-cube-cbrt0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 
(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))