Average Error: 0.2 → 0.0
Time: 19.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(b \cdot b + \left(\left(a \cdot a + a\right) + -3 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 4 - \left(1 - {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(b \cdot b + \left(\left(a \cdot a + a\right) + -3 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 4 - \left(1 - {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)
double f(double a, double b) {
        double r8803492 = a;
        double r8803493 = r8803492 * r8803492;
        double r8803494 = b;
        double r8803495 = r8803494 * r8803494;
        double r8803496 = r8803493 + r8803495;
        double r8803497 = 2.0;
        double r8803498 = pow(r8803496, r8803497);
        double r8803499 = 4.0;
        double r8803500 = 1.0;
        double r8803501 = r8803500 + r8803492;
        double r8803502 = r8803493 * r8803501;
        double r8803503 = 3.0;
        double r8803504 = r8803503 * r8803492;
        double r8803505 = r8803500 - r8803504;
        double r8803506 = r8803495 * r8803505;
        double r8803507 = r8803502 + r8803506;
        double r8803508 = r8803499 * r8803507;
        double r8803509 = r8803498 + r8803508;
        double r8803510 = r8803509 - r8803500;
        return r8803510;
}
↓
double f(double a, double b) {
        double r8803511 = b;
        double r8803512 = r8803511 * r8803511;
        double r8803513 = a;
        double r8803514 = r8803513 * r8803513;
        double r8803515 = r8803514 + r8803513;
        double r8803516 = -3.0;
        double r8803517 = r8803516 * r8803512;
        double r8803518 = r8803515 + r8803517;
        double r8803519 = r8803518 * r8803513;
        double r8803520 = r8803512 + r8803519;
        double r8803521 = 4.0;
        double r8803522 = r8803520 * r8803521;
        double r8803523 = 1.0;
        double r8803524 = r8803512 + r8803514;
        double r8803525 = sqrt(r8803524);
        double r8803526 = pow(r8803525, r8803521);
        double r8803527 = r8803523 - r8803526;
        double r8803528 = r8803522 - r8803527;
        return r8803528;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 
(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))