Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 21.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r4067220 = a;
        double r4067221 = r4067220 * r4067220;
        double r4067222 = b;
        double r4067223 = r4067222 * r4067222;
        double r4067224 = r4067221 + r4067223;
        double r4067225 = 2.0;
        double r4067226 = pow(r4067224, r4067225);
        double r4067227 = 4.0;
        double r4067228 = 1.0;
        double r4067229 = r4067228 + r4067220;
        double r4067230 = r4067221 * r4067229;
        double r4067231 = 3.0;
        double r4067232 = r4067231 * r4067220;
        double r4067233 = r4067228 - r4067232;
        double r4067234 = r4067223 * r4067233;
        double r4067235 = r4067230 + r4067234;
        double r4067236 = r4067227 * r4067235;
        double r4067237 = r4067226 + r4067236;
        double r4067238 = r4067237 - r4067228;
        return r4067238;
}

↓

double f(double a, double b) {
        double r4067239 = b;
        double r4067240 = r4067239 * r4067239;
        double r4067241 = 4.0;
        double r4067242 = a;
        double r4067243 = 12.0;
        double r4067244 = r4067242 * r4067243;
        double r4067245 = r4067241 - r4067244;
        double r4067246 = r4067240 * r4067245;
        double r4067247 = r4067242 * r4067242;
        double r4067248 = r4067241 * r4067247;
        double r4067249 = r4067248 * r4067242;
        double r4067250 = r4067247 + r4067240;
        double r4067251 = sqrt(r4067250);
        double r4067252 = pow(r4067251, r4067241);
        double r4067253 = 1.0;
        double r4067254 = r4067252 - r4067253;
        double r4067255 = r4067249 + r4067254;
        double r4067256 = r4067248 + r4067255;
        double r4067257 = r4067246 + r4067256;
        return r4067257;
}

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

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied associate-*r* 0.1

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

7. Applied add-sqr-sqrt 0.1

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

8. Applied pow3 0.1

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

10. Applied pow1 0.1

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

11. Applied pow-prod-up 0.0

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

12. Simplified 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019135 
(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))