Average Error: 0.2 → 0.2
Time: 22.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\]
\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
↓
\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(1 - 3 \cdot 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(1 - 3 \cdot a\right)\right)} - 1
double f(double a, double b) {
        double r94399 = a;
        double r94400 = r94399 * r94399;
        double r94401 = b;
        double r94402 = r94401 * r94401;
        double r94403 = r94400 + r94402;
        double r94404 = 2.0;
        double r94405 = pow(r94403, r94404);
        double r94406 = 4.0;
        double r94407 = 1.0;
        double r94408 = r94407 + r94399;
        double r94409 = r94400 * r94408;
        double r94410 = 3.0;
        double r94411 = r94410 * r94399;
        double r94412 = r94407 - r94411;
        double r94413 = r94402 * r94412;
        double r94414 = r94409 + r94413;
        double r94415 = r94406 * r94414;
        double r94416 = r94405 + r94415;
        double r94417 = r94416 - r94407;
        return r94417;
}
↓
double f(double a, double b) {
        double r94418 = a;
        double r94419 = r94418 * r94418;
        double r94420 = b;
        double r94421 = r94420 * r94420;
        double r94422 = r94419 + r94421;
        double r94423 = 2.0;
        double r94424 = pow(r94422, r94423);
        double r94425 = 4.0;
        double r94426 = 1.0;
        double r94427 = r94426 + r94418;
        double r94428 = r94419 * r94427;
        double r94429 = 3.0;
        double r94430 = r94429 * r94418;
        double r94431 = r94426 - r94430;
        double r94432 = r94421 * r94431;
        double r94433 = r94428 + r94432;
        double r94434 = r94425 * r94433;
        double r94435 = r94424 + r94434;
        double r94436 = sqrt(r94435);
        double r94437 = r94436 * r94436;
        double r94438 = r94437 - r94426;
        return r94438;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))