Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.1

Time: 27.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r49676262 = a;
        double r49676263 = r49676262 * r49676262;
        double r49676264 = b;
        double r49676265 = r49676264 * r49676264;
        double r49676266 = r49676263 + r49676265;
        double r49676267 = 2.0;
        double r49676268 = pow(r49676266, r49676267);
        double r49676269 = 4.0;
        double r49676270 = r49676269 * r49676265;
        double r49676271 = r49676268 + r49676270;
        double r49676272 = 1.0;
        double r49676273 = r49676271 - r49676272;
        return r49676273;
}

↓

double f(double a, double b) {
        double r49676274 = 4.0;
        double r49676275 = b;
        double r49676276 = r49676275 * r49676275;
        double r49676277 = r49676274 * r49676276;
        double r49676278 = a;
        double r49676279 = r49676278 * r49676278;
        double r49676280 = 2.0;
        double r49676281 = r49676279 * r49676280;
        double r49676282 = fma(r49676275, r49676275, r49676281);
        double r49676283 = r49676282 * r49676275;
        double r49676284 = pow(r49676278, r49676274);
        double r49676285 = fma(r49676275, r49676283, r49676284);
        double r49676286 = r49676277 + r49676285;
        double r49676287 = 1.0;
        double r49676288 = r49676286 - r49676287;
        return r49676288;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Taylor expanded around 0 0.0

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

3. Simplified 0.2

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-prod-up 0.2

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

8. Applied pow1 0.2

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

9. Applied pow1 0.2

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

10. Applied pow-sqr 0.2

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

11. Applied pow-prod-up 0.1

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

12. Simplified 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))