Average Error: 0.2 → 0.0
Time: 28.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1\]
\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(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1
double f(double a, double b) {
        double r3699590 = a;
        double r3699591 = r3699590 * r3699590;
        double r3699592 = b;
        double r3699593 = r3699592 * r3699592;
        double r3699594 = r3699591 + r3699593;
        double r3699595 = 2.0;
        double r3699596 = pow(r3699594, r3699595);
        double r3699597 = 4.0;
        double r3699598 = 1.0;
        double r3699599 = r3699598 + r3699590;
        double r3699600 = r3699591 * r3699599;
        double r3699601 = 3.0;
        double r3699602 = r3699601 * r3699590;
        double r3699603 = r3699598 - r3699602;
        double r3699604 = r3699593 * r3699603;
        double r3699605 = r3699600 + r3699604;
        double r3699606 = r3699597 * r3699605;
        double r3699607 = r3699596 + r3699606;
        double r3699608 = r3699607 - r3699598;
        return r3699608;
}
↓
double f(double a, double b) {
        double r3699609 = b;
        double r3699610 = r3699609 * r3699609;
        double r3699611 = a;
        double r3699612 = r3699611 * r3699611;
        double r3699613 = r3699610 + r3699612;
        double r3699614 = sqrt(r3699613);
        double r3699615 = 4.0;
        double r3699616 = pow(r3699614, r3699615);
        double r3699617 = -3.0;
        double r3699618 = r3699617 * r3699610;
        double r3699619 = r3699611 + r3699612;
        double r3699620 = r3699618 + r3699619;
        double r3699621 = r3699620 * r3699611;
        double r3699622 = r3699621 + r3699610;
        double r3699623 = r3699622 * r3699615;
        double r3699624 = r3699616 + r3699623;
        double r3699625 = -1.0;
        double r3699626 = r3699624 + r3699625;
        return r3699626;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 
(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))