Average Error: 0.2 → 0.0
Time: 26.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\right) - 1\]
\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(-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\right) - 1
double f(double a, double b) {
        double r8049713 = a;
        double r8049714 = r8049713 * r8049713;
        double r8049715 = b;
        double r8049716 = r8049715 * r8049715;
        double r8049717 = r8049714 + r8049716;
        double r8049718 = 2.0;
        double r8049719 = pow(r8049717, r8049718);
        double r8049720 = 4.0;
        double r8049721 = 1.0;
        double r8049722 = r8049721 + r8049713;
        double r8049723 = r8049714 * r8049722;
        double r8049724 = 3.0;
        double r8049725 = r8049724 * r8049713;
        double r8049726 = r8049721 - r8049725;
        double r8049727 = r8049716 * r8049726;
        double r8049728 = r8049723 + r8049727;
        double r8049729 = r8049720 * r8049728;
        double r8049730 = r8049719 + r8049729;
        double r8049731 = r8049730 - r8049721;
        return r8049731;
}
↓
double f(double a, double b) {
        double r8049732 = a;
        double r8049733 = r8049732 * r8049732;
        double r8049734 = r8049733 * r8049732;
        double r8049735 = b;
        double r8049736 = r8049735 * r8049735;
        double r8049737 = r8049736 + r8049733;
        double r8049738 = r8049734 + r8049737;
        double r8049739 = 4.0;
        double r8049740 = r8049738 * r8049739;
        double r8049741 = -12.0;
        double r8049742 = r8049736 * r8049732;
        double r8049743 = r8049741 * r8049742;
        double r8049744 = sqrt(r8049737);
        double r8049745 = pow(r8049744, r8049739);
        double r8049746 = r8049743 + r8049745;
        double r8049747 = r8049740 + r8049746;
        double r8049748 = 1.0;
        double r8049749 = r8049747 - r8049748;
        return r8049749;
}

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. Using strategy rm

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 
(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))