Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 20.6s

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 r4426248 = a;
        double r4426249 = r4426248 * r4426248;
        double r4426250 = b;
        double r4426251 = r4426250 * r4426250;
        double r4426252 = r4426249 + r4426251;
        double r4426253 = 2.0;
        double r4426254 = pow(r4426252, r4426253);
        double r4426255 = 4.0;
        double r4426256 = 1.0;
        double r4426257 = r4426256 + r4426248;
        double r4426258 = r4426249 * r4426257;
        double r4426259 = 3.0;
        double r4426260 = r4426259 * r4426248;
        double r4426261 = r4426256 - r4426260;
        double r4426262 = r4426251 * r4426261;
        double r4426263 = r4426258 + r4426262;
        double r4426264 = r4426255 * r4426263;
        double r4426265 = r4426254 + r4426264;
        double r4426266 = r4426265 - r4426256;
        return r4426266;
}

↓

double f(double a, double b) {
        double r4426267 = a;
        double r4426268 = r4426267 * r4426267;
        double r4426269 = r4426268 * r4426267;
        double r4426270 = b;
        double r4426271 = r4426270 * r4426270;
        double r4426272 = r4426271 + r4426268;
        double r4426273 = r4426269 + r4426272;
        double r4426274 = 4.0;
        double r4426275 = r4426273 * r4426274;
        double r4426276 = -12.0;
        double r4426277 = r4426271 * r4426267;
        double r4426278 = r4426276 * r4426277;
        double r4426279 = sqrt(r4426272);
        double r4426280 = pow(r4426279, r4426274);
        double r4426281 = r4426278 + r4426280;
        double r4426282 = r4426275 + r4426281;
        double r4426283 = 1.0;
        double r4426284 = r4426282 - r4426283;
        return r4426284;
}

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 2019152 
(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))