Average Error: 0.2 → 0.0
Time: 25.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\right) - 1\]
\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 r8093007 = a;
        double r8093008 = r8093007 * r8093007;
        double r8093009 = b;
        double r8093010 = r8093009 * r8093009;
        double r8093011 = r8093008 + r8093010;
        double r8093012 = 2.0;
        double r8093013 = pow(r8093011, r8093012);
        double r8093014 = 4.0;
        double r8093015 = 1.0;
        double r8093016 = r8093015 + r8093007;
        double r8093017 = r8093008 * r8093016;
        double r8093018 = 3.0;
        double r8093019 = r8093018 * r8093007;
        double r8093020 = r8093015 - r8093019;
        double r8093021 = r8093010 * r8093020;
        double r8093022 = r8093017 + r8093021;
        double r8093023 = r8093014 * r8093022;
        double r8093024 = r8093013 + r8093023;
        double r8093025 = r8093024 - r8093015;
        return r8093025;
}
↓
double f(double a, double b) {
        double r8093026 = a;
        double r8093027 = r8093026 * r8093026;
        double r8093028 = r8093027 * r8093026;
        double r8093029 = b;
        double r8093030 = r8093029 * r8093029;
        double r8093031 = r8093030 + r8093027;
        double r8093032 = r8093028 + r8093031;
        double r8093033 = 4.0;
        double r8093034 = r8093032 * r8093033;
        double r8093035 = -12.0;
        double r8093036 = r8093030 * r8093026;
        double r8093037 = r8093035 * r8093036;
        double r8093038 = sqrt(r8093031);
        double r8093039 = pow(r8093038, r8093033);
        double r8093040 = r8093037 + r8093039;
        double r8093041 = r8093034 + r8093040;
        double r8093042 = 1.0;
        double r8093043 = r8093041 - r8093042;
        return r8093043;
}

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

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  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 2019146 
(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))