Average Error: 0.2 → 0.0
Time: 21.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(b \cdot b + \left(\left(a \cdot a + a\right) + -3 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 4 - \left(1 - {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\]
\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 + \left(\left(a \cdot a + a\right) + -3 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot 4 - \left(1 - {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)
double f(double a, double b) {
        double r3898198 = a;
        double r3898199 = r3898198 * r3898198;
        double r3898200 = b;
        double r3898201 = r3898200 * r3898200;
        double r3898202 = r3898199 + r3898201;
        double r3898203 = 2.0;
        double r3898204 = pow(r3898202, r3898203);
        double r3898205 = 4.0;
        double r3898206 = 1.0;
        double r3898207 = r3898206 + r3898198;
        double r3898208 = r3898199 * r3898207;
        double r3898209 = 3.0;
        double r3898210 = r3898209 * r3898198;
        double r3898211 = r3898206 - r3898210;
        double r3898212 = r3898201 * r3898211;
        double r3898213 = r3898208 + r3898212;
        double r3898214 = r3898205 * r3898213;
        double r3898215 = r3898204 + r3898214;
        double r3898216 = r3898215 - r3898206;
        return r3898216;
}
↓
double f(double a, double b) {
        double r3898217 = b;
        double r3898218 = r3898217 * r3898217;
        double r3898219 = a;
        double r3898220 = r3898219 * r3898219;
        double r3898221 = r3898220 + r3898219;
        double r3898222 = -3.0;
        double r3898223 = r3898222 * r3898218;
        double r3898224 = r3898221 + r3898223;
        double r3898225 = r3898224 * r3898219;
        double r3898226 = r3898218 + r3898225;
        double r3898227 = 4.0;
        double r3898228 = r3898226 * r3898227;
        double r3898229 = 1.0;
        double r3898230 = r3898218 + r3898220;
        double r3898231 = sqrt(r3898230);
        double r3898232 = pow(r3898231, r3898227);
        double r3898233 = r3898229 - r3898232;
        double r3898234 = r3898228 - r3898233;
        return r3898234;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 
(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))