Average Error: 0.2 → 0.1
Time: 25.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1\]
\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
↓
\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1
double f(double a, double b) {
        double r7877476 = a;
        double r7877477 = r7877476 * r7877476;
        double r7877478 = b;
        double r7877479 = r7877478 * r7877478;
        double r7877480 = r7877477 + r7877479;
        double r7877481 = 2.0;
        double r7877482 = pow(r7877480, r7877481);
        double r7877483 = 4.0;
        double r7877484 = 1.0;
        double r7877485 = r7877484 + r7877476;
        double r7877486 = r7877477 * r7877485;
        double r7877487 = 3.0;
        double r7877488 = r7877487 * r7877476;
        double r7877489 = r7877484 - r7877488;
        double r7877490 = r7877479 * r7877489;
        double r7877491 = r7877486 + r7877490;
        double r7877492 = r7877483 * r7877491;
        double r7877493 = r7877482 + r7877492;
        double r7877494 = r7877493 - r7877484;
        return r7877494;
}
↓
double f(double a, double b) {
        double r7877495 = a;
        double r7877496 = r7877495 * r7877495;
        double r7877497 = r7877496 * r7877495;
        double r7877498 = b;
        double r7877499 = r7877498 * r7877498;
        double r7877500 = r7877499 + r7877496;
        double r7877501 = r7877497 + r7877500;
        double r7877502 = 4.0;
        double r7877503 = r7877501 * r7877502;
        double r7877504 = r7877499 * r7877495;
        double r7877505 = -12.0;
        double r7877506 = r7877504 * r7877505;
        double r7877507 = sqrt(r7877500);
        double r7877508 = 3.0;
        double r7877509 = pow(r7877507, r7877508);
        double r7877510 = r7877509 * r7877507;
        double r7877511 = r7877506 + r7877510;
        double r7877512 = r7877503 + r7877511;
        double r7877513 = 1.0;
        double r7877514 = r7877512 - r7877513;
        return r7877514;
}

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

    \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  5. Applied associate-*l*0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(a \cdot a + b \cdot b\right)\right)} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  8. Applied cube-unmult0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

  9. Final simplification0.1

    \[\leadsto \left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{3} \cdot \sqrt{b \cdot b + a \cdot a}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019151 
(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))