Average Error: 0.2 → 0.0
Time: 19.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - \left(1 - \left(\left(\left(a + 3\right) \cdot \left(b \cdot b\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4\right)\]
\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(3 + a\right)\right)\right) - 1
↓
{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - \left(1 - \left(\left(\left(a + 3\right) \cdot \left(b \cdot b\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4\right)
double f(double a, double b) {
        double r10460186 = a;
        double r10460187 = r10460186 * r10460186;
        double r10460188 = b;
        double r10460189 = r10460188 * r10460188;
        double r10460190 = r10460187 + r10460189;
        double r10460191 = 2.0;
        double r10460192 = pow(r10460190, r10460191);
        double r10460193 = 4.0;
        double r10460194 = 1.0;
        double r10460195 = r10460194 - r10460186;
        double r10460196 = r10460187 * r10460195;
        double r10460197 = 3.0;
        double r10460198 = r10460197 + r10460186;
        double r10460199 = r10460189 * r10460198;
        double r10460200 = r10460196 + r10460199;
        double r10460201 = r10460193 * r10460200;
        double r10460202 = r10460192 + r10460201;
        double r10460203 = r10460202 - r10460194;
        return r10460203;
}
↓
double f(double a, double b) {
        double r10460204 = a;
        double r10460205 = r10460204 * r10460204;
        double r10460206 = b;
        double r10460207 = r10460206 * r10460206;
        double r10460208 = r10460205 + r10460207;
        double r10460209 = sqrt(r10460208);
        double r10460210 = 4.0;
        double r10460211 = pow(r10460209, r10460210);
        double r10460212 = 1.0;
        double r10460213 = 3.0;
        double r10460214 = r10460204 + r10460213;
        double r10460215 = r10460214 * r10460207;
        double r10460216 = r10460215 + r10460205;
        double r10460217 = r10460204 * r10460205;
        double r10460218 = r10460216 - r10460217;
        double r10460219 = r10460218 * r10460210;
        double r10460220 = r10460212 - r10460219;
        double r10460221 = r10460211 - r10460220;
        return r10460221;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))