Average Error: 0.2 → 0.0
Time: 52.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\frac{\left(1 - a \cdot a\right) \cdot \left(a \cdot a\right)}{a + 1} + \left(3 + a\right) \cdot \left(b \cdot b\right)\right) + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left(4 \cdot \left(\frac{\left(1 - a \cdot a\right) \cdot \left(a \cdot a\right)}{a + 1} + \left(3 + a\right) \cdot \left(b \cdot b\right)\right) + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r66211512 = a;
        double r66211513 = r66211512 * r66211512;
        double r66211514 = b;
        double r66211515 = r66211514 * r66211514;
        double r66211516 = r66211513 + r66211515;
        double r66211517 = 2.0;
        double r66211518 = pow(r66211516, r66211517);
        double r66211519 = 4.0;
        double r66211520 = 1.0;
        double r66211521 = r66211520 - r66211512;
        double r66211522 = r66211513 * r66211521;
        double r66211523 = 3.0;
        double r66211524 = r66211523 + r66211512;
        double r66211525 = r66211515 * r66211524;
        double r66211526 = r66211522 + r66211525;
        double r66211527 = r66211519 * r66211526;
        double r66211528 = r66211518 + r66211527;
        double r66211529 = r66211528 - r66211520;
        return r66211529;
}
↓
double f(double a, double b) {
        double r66211530 = 4.0;
        double r66211531 = 1.0;
        double r66211532 = a;
        double r66211533 = r66211532 * r66211532;
        double r66211534 = r66211531 - r66211533;
        double r66211535 = r66211534 * r66211533;
        double r66211536 = r66211532 + r66211531;
        double r66211537 = r66211535 / r66211536;
        double r66211538 = 3.0;
        double r66211539 = r66211538 + r66211532;
        double r66211540 = b;
        double r66211541 = r66211540 * r66211540;
        double r66211542 = r66211539 * r66211541;
        double r66211543 = r66211537 + r66211542;
        double r66211544 = r66211530 * r66211543;
        double r66211545 = r66211533 + r66211541;
        double r66211546 = sqrt(r66211545);
        double r66211547 = pow(r66211546, r66211530);
        double r66211548 = r66211544 + r66211547;
        double r66211549 = r66211548 - r66211531;
        return r66211549;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied unpow-prod-down0.2

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

  6. Applied pow-sqr0.0

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

  7. Simplified0.0

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

  9. Applied flip--0.0

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

  10. Applied associate-*r/0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019112 
(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))