Average Error: 0.2 → 0.0
Time: 46.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right) + a \cdot \left(a \cdot a + a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
double f(double a, double b) {
        double r62584206 = a;
        double r62584207 = r62584206 * r62584206;
        double r62584208 = b;
        double r62584209 = r62584208 * r62584208;
        double r62584210 = r62584207 + r62584209;
        double r62584211 = 2.0;
        double r62584212 = pow(r62584210, r62584211);
        double r62584213 = 4.0;
        double r62584214 = 1.0;
        double r62584215 = r62584214 + r62584206;
        double r62584216 = r62584207 * r62584215;
        double r62584217 = 3.0;
        double r62584218 = r62584217 * r62584206;
        double r62584219 = r62584214 - r62584218;
        double r62584220 = r62584209 * r62584219;
        double r62584221 = r62584216 + r62584220;
        double r62584222 = r62584213 * r62584221;
        double r62584223 = r62584212 + r62584222;
        double r62584224 = r62584223 - r62584214;
        return r62584224;
}
↓
double f(double a, double b) {
        double r62584225 = b;
        double r62584226 = 1.0;
        double r62584227 = a;
        double r62584228 = 3.0;
        double r62584229 = r62584227 * r62584228;
        double r62584230 = r62584226 - r62584229;
        double r62584231 = r62584225 * r62584230;
        double r62584232 = r62584225 * r62584231;
        double r62584233 = r62584227 * r62584227;
        double r62584234 = r62584233 + r62584227;
        double r62584235 = r62584227 * r62584234;
        double r62584236 = r62584232 + r62584235;
        double r62584237 = 4.0;
        double r62584238 = r62584236 * r62584237;
        double r62584239 = r62584225 * r62584225;
        double r62584240 = r62584233 + r62584239;
        double r62584241 = sqrt(r62584240);
        double r62584242 = pow(r62584241, r62584237);
        double r62584243 = r62584226 - r62584242;
        double r62584244 = r62584238 - r62584243;
        return r62584244;
}


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

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(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))