Average Error: 0.2 → 0.2
Time: 18.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} - 1\]
\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
↓
\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} - 1
double f(double a, double b) {
        double r8334547 = a;
        double r8334548 = r8334547 * r8334547;
        double r8334549 = b;
        double r8334550 = r8334549 * r8334549;
        double r8334551 = r8334548 + r8334550;
        double r8334552 = 2.0;
        double r8334553 = pow(r8334551, r8334552);
        double r8334554 = 4.0;
        double r8334555 = 1.0;
        double r8334556 = r8334555 - r8334547;
        double r8334557 = r8334548 * r8334556;
        double r8334558 = 3.0;
        double r8334559 = r8334558 + r8334547;
        double r8334560 = r8334550 * r8334559;
        double r8334561 = r8334557 + r8334560;
        double r8334562 = r8334554 * r8334561;
        double r8334563 = r8334553 + r8334562;
        double r8334564 = r8334563 - r8334555;
        return r8334564;
}
↓
double f(double a, double b) {
        double r8334565 = a;
        double r8334566 = r8334565 * r8334565;
        double r8334567 = b;
        double r8334568 = r8334567 * r8334567;
        double r8334569 = r8334566 + r8334568;
        double r8334570 = 2.0;
        double r8334571 = pow(r8334569, r8334570);
        double r8334572 = 3.0;
        double r8334573 = r8334565 + r8334572;
        double r8334574 = r8334573 * r8334568;
        double r8334575 = 1.0;
        double r8334576 = r8334575 - r8334565;
        double r8334577 = r8334566 * r8334576;
        double r8334578 = r8334574 + r8334577;
        double r8334579 = 4.0;
        double r8334580 = r8334578 * r8334579;
        double r8334581 = r8334571 + r8334580;
        double r8334582 = sqrt(r8334581);
        double r8334583 = r8334582 * r8334582;
        double r8334584 = r8334583 - r8334575;
        return r8334584;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))