Average Error: 0.2 → 0.0
Time: 1.0m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + \left(\left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4 - 1\right)\]
\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
↓
{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + \left(\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\right)
double f(double a, double b) {
        double r71701715 = a;
        double r71701716 = r71701715 * r71701715;
        double r71701717 = b;
        double r71701718 = r71701717 * r71701717;
        double r71701719 = r71701716 + r71701718;
        double r71701720 = 2.0;
        double r71701721 = pow(r71701719, r71701720);
        double r71701722 = 4.0;
        double r71701723 = 1.0;
        double r71701724 = r71701723 - r71701715;
        double r71701725 = r71701716 * r71701724;
        double r71701726 = 3.0;
        double r71701727 = r71701726 + r71701715;
        double r71701728 = r71701718 * r71701727;
        double r71701729 = r71701725 + r71701728;
        double r71701730 = r71701722 * r71701729;
        double r71701731 = r71701721 + r71701730;
        double r71701732 = r71701731 - r71701723;
        return r71701732;
}
↓
double f(double a, double b) {
        double r71701733 = a;
        double r71701734 = r71701733 * r71701733;
        double r71701735 = b;
        double r71701736 = r71701735 * r71701735;
        double r71701737 = r71701734 + r71701736;
        double r71701738 = sqrt(r71701737);
        double r71701739 = 4.0;
        double r71701740 = pow(r71701738, r71701739);
        double r71701741 = 3.0;
        double r71701742 = r71701733 + r71701741;
        double r71701743 = r71701742 * r71701736;
        double r71701744 = 1.0;
        double r71701745 = r71701744 - r71701733;
        double r71701746 = r71701734 * r71701745;
        double r71701747 = r71701743 + r71701746;
        double r71701748 = r71701747 * r71701739;
        double r71701749 = r71701748 - r71701744;
        double r71701750 = r71701740 + r71701749;
        return r71701750;
}

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 associate--l+0.2

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied associate-*l*0.1

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

  9. Applied add-sqr-sqrt0.2

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

  10. Applied cube-unmult0.1

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

  11. Applied pow10.1

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

  12. Applied pow-prod-up0.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))