Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 20.2s

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 r9412076 = a;
        double r9412077 = r9412076 * r9412076;
        double r9412078 = b;
        double r9412079 = r9412078 * r9412078;
        double r9412080 = r9412077 + r9412079;
        double r9412081 = 2.0;
        double r9412082 = pow(r9412080, r9412081);
        double r9412083 = 4.0;
        double r9412084 = 1.0;
        double r9412085 = r9412084 - r9412076;
        double r9412086 = r9412077 * r9412085;
        double r9412087 = 3.0;
        double r9412088 = r9412087 + r9412076;
        double r9412089 = r9412079 * r9412088;
        double r9412090 = r9412086 + r9412089;
        double r9412091 = r9412083 * r9412090;
        double r9412092 = r9412082 + r9412091;
        double r9412093 = r9412092 - r9412084;
        return r9412093;
}

↓

double f(double a, double b) {
        double r9412094 = 4.0;
        double r9412095 = 1.0;
        double r9412096 = a;
        double r9412097 = r9412095 - r9412096;
        double r9412098 = r9412096 * r9412096;
        double r9412099 = r9412097 * r9412098;
        double r9412100 = 3.0;
        double r9412101 = r9412100 + r9412096;
        double r9412102 = b;
        double r9412103 = r9412102 * r9412102;
        double r9412104 = r9412101 * r9412103;
        double r9412105 = r9412099 + r9412104;
        double r9412106 = r9412094 * r9412105;
        double r9412107 = pow(r9412096, r9412094);
        double r9412108 = 2.0;
        double r9412109 = r9412098 * r9412108;
        double r9412110 = r9412103 + r9412109;
        double r9412111 = r9412110 * r9412103;
        double r9412112 = r9412107 + r9412111;
        double r9412113 = r9412106 + r9412112;
        double r9412114 = r9412113 - r9412095;
        return r9412114;
}

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))