Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 28.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r6240155 = a;
        double r6240156 = r6240155 * r6240155;
        double r6240157 = b;
        double r6240158 = r6240157 * r6240157;
        double r6240159 = r6240156 + r6240158;
        double r6240160 = 2.0;
        double r6240161 = pow(r6240159, r6240160);
        double r6240162 = 4.0;
        double r6240163 = 1.0;
        double r6240164 = r6240163 + r6240155;
        double r6240165 = r6240156 * r6240164;
        double r6240166 = 3.0;
        double r6240167 = r6240166 * r6240155;
        double r6240168 = r6240163 - r6240167;
        double r6240169 = r6240158 * r6240168;
        double r6240170 = r6240165 + r6240169;
        double r6240171 = r6240162 * r6240170;
        double r6240172 = r6240161 + r6240171;
        double r6240173 = r6240172 - r6240163;
        return r6240173;
}

↓

double f(double a, double b) {
        double r6240174 = a;
        double r6240175 = r6240174 * r6240174;
        double r6240176 = r6240175 * r6240174;
        double r6240177 = b;
        double r6240178 = r6240177 * r6240177;
        double r6240179 = r6240178 + r6240175;
        double r6240180 = r6240176 + r6240179;
        double r6240181 = 4.0;
        double r6240182 = r6240180 * r6240181;
        double r6240183 = -12.0;
        double r6240184 = r6240178 * r6240174;
        double r6240185 = r6240183 * r6240184;
        double r6240186 = sqrt(r6240179);
        double r6240187 = pow(r6240186, r6240181);
        double r6240188 = r6240185 + r6240187;
        double r6240189 = r6240182 + r6240188;
        double r6240190 = 1.0;
        double r6240191 = r6240189 - r6240190;
        return r6240191;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

4. Applied add-sqr-sqrt 0.2

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

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 pow1 0.1

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

8. Applied add-sqr-sqrt 0.1

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

9. Applied pow3 0.1

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

10. Applied pow-prod-up 0.0

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

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019144 
(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))