Average Error: 0.2 → 0.0
Time: 17.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4\]
\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(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4
double f(double a, double b) {
        double r8127585 = a;
        double r8127586 = r8127585 * r8127585;
        double r8127587 = b;
        double r8127588 = r8127587 * r8127587;
        double r8127589 = r8127586 + r8127588;
        double r8127590 = 2.0;
        double r8127591 = pow(r8127589, r8127590);
        double r8127592 = 4.0;
        double r8127593 = 1.0;
        double r8127594 = r8127593 + r8127585;
        double r8127595 = r8127586 * r8127594;
        double r8127596 = 3.0;
        double r8127597 = r8127596 * r8127585;
        double r8127598 = r8127593 - r8127597;
        double r8127599 = r8127588 * r8127598;
        double r8127600 = r8127595 + r8127599;
        double r8127601 = r8127592 * r8127600;
        double r8127602 = r8127591 + r8127601;
        double r8127603 = r8127602 - r8127593;
        return r8127603;
}
↓
double f(double a, double b) {
        double r8127604 = 4.0;
        double r8127605 = a;
        double r8127606 = r8127605 * r8127605;
        double r8127607 = b;
        double r8127608 = r8127607 * r8127607;
        double r8127609 = r8127606 + r8127608;
        double r8127610 = -3.0;
        double r8127611 = r8127605 * r8127608;
        double r8127612 = r8127610 * r8127611;
        double r8127613 = r8127609 + r8127612;
        double r8127614 = r8127604 * r8127613;
        double r8127615 = -1.0;
        double r8127616 = sqrt(r8127609);
        double r8127617 = pow(r8127616, r8127604);
        double r8127618 = r8127615 + r8127617;
        double r8127619 = r8127614 + r8127618;
        double r8127620 = cbrt(r8127605);
        double r8127621 = r8127620 * r8127620;
        double r8127622 = r8127621 * r8127606;
        double r8127623 = r8127620 * r8127622;
        double r8127624 = r8127623 * r8127604;
        double r8127625 = r8127619 + r8127624;
        return r8127625;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  14. Applied add-cube-cbrt0.0

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

  15. Applied associate-*r*0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(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))