Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 20.8s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r9864214 = a;
        double r9864215 = r9864214 * r9864214;
        double r9864216 = b;
        double r9864217 = r9864216 * r9864216;
        double r9864218 = r9864215 + r9864217;
        double r9864219 = 2.0;
        double r9864220 = pow(r9864218, r9864219);
        double r9864221 = 4.0;
        double r9864222 = 1.0;
        double r9864223 = r9864222 - r9864214;
        double r9864224 = r9864215 * r9864223;
        double r9864225 = 3.0;
        double r9864226 = r9864225 + r9864214;
        double r9864227 = r9864217 * r9864226;
        double r9864228 = r9864224 + r9864227;
        double r9864229 = r9864221 * r9864228;
        double r9864230 = r9864220 + r9864229;
        double r9864231 = r9864230 - r9864222;
        return r9864231;
}

↓

double f(double a, double b) {
        double r9864232 = 4.0;
        double r9864233 = 1.0;
        double r9864234 = a;
        double r9864235 = r9864233 - r9864234;
        double r9864236 = r9864234 * r9864234;
        double r9864237 = r9864235 * r9864236;
        double r9864238 = 3.0;
        double r9864239 = r9864238 + r9864234;
        double r9864240 = b;
        double r9864241 = r9864240 * r9864240;
        double r9864242 = r9864239 * r9864241;
        double r9864243 = r9864237 + r9864242;
        double r9864244 = r9864232 * r9864243;
        double r9864245 = pow(r9864234, r9864232);
        double r9864246 = 2.0;
        double r9864247 = r9864236 * r9864246;
        double r9864248 = r9864241 + r9864247;
        double r9864249 = r9864248 * r9864241;
        double r9864250 = r9864245 + r9864249;
        double r9864251 = r9864244 + r9864250;
        double r9864252 = r9864251 - r9864233;
        return r9864252;
}

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-prod-up 0.2

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

8. Applied pow1 0.2

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

9. Applied pow1 0.2

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

10. Applied pow-prod-up 0.2

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

11. Applied pow-prod-up 0.1

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

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. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019162 
(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))