Average Error: 0.2 → 0.0
Time: 58.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right) + a \cdot \left(a \cdot a + a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
double f(double a, double b) {
        double r20964355 = a;
        double r20964356 = r20964355 * r20964355;
        double r20964357 = b;
        double r20964358 = r20964357 * r20964357;
        double r20964359 = r20964356 + r20964358;
        double r20964360 = 2.0;
        double r20964361 = pow(r20964359, r20964360);
        double r20964362 = 4.0;
        double r20964363 = 1.0;
        double r20964364 = r20964363 + r20964355;
        double r20964365 = r20964356 * r20964364;
        double r20964366 = 3.0;
        double r20964367 = r20964366 * r20964355;
        double r20964368 = r20964363 - r20964367;
        double r20964369 = r20964358 * r20964368;
        double r20964370 = r20964365 + r20964369;
        double r20964371 = r20964362 * r20964370;
        double r20964372 = r20964361 + r20964371;
        double r20964373 = r20964372 - r20964363;
        return r20964373;
}
↓
double f(double a, double b) {
        double r20964374 = b;
        double r20964375 = 1.0;
        double r20964376 = a;
        double r20964377 = 3.0;
        double r20964378 = r20964376 * r20964377;
        double r20964379 = r20964375 - r20964378;
        double r20964380 = r20964374 * r20964379;
        double r20964381 = r20964374 * r20964380;
        double r20964382 = r20964376 * r20964376;
        double r20964383 = r20964382 + r20964376;
        double r20964384 = r20964376 * r20964383;
        double r20964385 = r20964381 + r20964384;
        double r20964386 = 4.0;
        double r20964387 = r20964385 * r20964386;
        double r20964388 = r20964374 * r20964374;
        double r20964389 = r20964382 + r20964388;
        double r20964390 = sqrt(r20964389);
        double r20964391 = pow(r20964390, r20964386);
        double r20964392 = r20964375 - r20964391;
        double r20964393 = r20964387 - r20964392;
        return r20964393;
}


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

Error

Bits error versus a

Bits error versus b

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied pow10.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied pow30.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(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))