Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 21.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r6847998 = a;
        double r6847999 = r6847998 * r6847998;
        double r6848000 = b;
        double r6848001 = r6848000 * r6848000;
        double r6848002 = r6847999 + r6848001;
        double r6848003 = 2.0;
        double r6848004 = pow(r6848002, r6848003);
        double r6848005 = 4.0;
        double r6848006 = 1.0;
        double r6848007 = r6848006 - r6847998;
        double r6848008 = r6847999 * r6848007;
        double r6848009 = 3.0;
        double r6848010 = r6848009 + r6847998;
        double r6848011 = r6848001 * r6848010;
        double r6848012 = r6848008 + r6848011;
        double r6848013 = r6848005 * r6848012;
        double r6848014 = r6848004 + r6848013;
        double r6848015 = r6848014 - r6848006;
        return r6848015;
}

↓

double f(double a, double b) {
        double r6848016 = 4.0;
        double r6848017 = a;
        double r6848018 = r6848017 * r6848017;
        double r6848019 = 1.0;
        double r6848020 = r6848019 - r6848017;
        double r6848021 = r6848018 * r6848020;
        double r6848022 = 3.0;
        double r6848023 = r6848017 + r6848022;
        double r6848024 = b;
        double r6848025 = r6848024 * r6848024;
        double r6848026 = r6848023 * r6848025;
        double r6848027 = r6848021 + r6848026;
        double r6848028 = r6848016 * r6848027;
        double r6848029 = pow(r6848017, r6848016);
        double r6848030 = 2.0;
        double r6848031 = r6848018 * r6848030;
        double r6848032 = r6848025 + r6848031;
        double r6848033 = r6848024 * r6848032;
        double r6848034 = r6848033 * r6848024;
        double r6848035 = r6848029 + r6848034;
        double r6848036 = r6848028 + r6848035;
        double r6848037 = r6848036 - r6848019;
        return r6848037;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Taylor expanded around 0 0.0

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

3. Simplified 0.2

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-sqr 0.2

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

8. Applied pow1 0.2

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

9. Applied pow1 0.2

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

10. Applied pow-sqr 0.2

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

11. Applied pow-sqr 0.1

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

12. Simplified 0.1

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

14. Applied associate-*l* 0.1

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

15. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(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))