Average Error: 0.2 → 0.0
Time: 44.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
double f(double a, double b) {
        double r42199503 = a;
        double r42199504 = r42199503 * r42199503;
        double r42199505 = b;
        double r42199506 = r42199505 * r42199505;
        double r42199507 = r42199504 + r42199506;
        double r42199508 = 2.0;
        double r42199509 = pow(r42199507, r42199508);
        double r42199510 = 4.0;
        double r42199511 = r42199510 * r42199506;
        double r42199512 = r42199509 + r42199511;
        double r42199513 = 1.0;
        double r42199514 = r42199512 - r42199513;
        return r42199514;
}
↓
double f(double a, double b) {
        double r42199515 = 4.0;
        double r42199516 = b;
        double r42199517 = r42199516 * r42199516;
        double r42199518 = r42199515 * r42199517;
        double r42199519 = cbrt(r42199518);
        double r42199520 = r42199519 * r42199519;
        double r42199521 = r42199519 * r42199520;
        double r42199522 = a;
        double r42199523 = r42199522 * r42199522;
        double r42199524 = r42199517 + r42199523;
        double r42199525 = sqrt(r42199524);
        double r42199526 = pow(r42199525, r42199515);
        double r42199527 = r42199521 + r42199526;
        double r42199528 = 1.0;
        double r42199529 = r42199527 - r42199528;
        return r42199529;
}


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

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  13. Applied add-cube-cbrt0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))