Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)}\right) \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\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({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)}\right) \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
double f(double a, double b) {
        double r394203 = a;
        double r394204 = r394203 * r394203;
        double r394205 = b;
        double r394206 = r394205 * r394205;
        double r394207 = r394204 + r394206;
        double r394208 = 2.0;
        double r394209 = pow(r394207, r394208);
        double r394210 = 4.0;
        double r394211 = 1.0;
        double r394212 = r394211 - r394203;
        double r394213 = r394204 * r394212;
        double r394214 = 3.0;
        double r394215 = r394214 + r394203;
        double r394216 = r394206 * r394215;
        double r394217 = r394213 + r394216;
        double r394218 = r394210 * r394217;
        double r394219 = r394209 + r394218;
        double r394220 = r394219 - r394211;
        return r394220;
}
↓
double f(double a, double b) {
        double r394221 = a;
        double r394222 = r394221 * r394221;
        double r394223 = b;
        double r394224 = r394223 * r394223;
        double r394225 = r394222 + r394224;
        double r394226 = 2.0;
        double r394227 = pow(r394225, r394226);
        double r394228 = 4.0;
        double r394229 = 1.0;
        double r394230 = r394229 - r394221;
        double r394231 = r394222 * r394230;
        double r394232 = cbrt(r394231);
        double r394233 = r394232 * r394232;
        double r394234 = r394233 * r394232;
        double r394235 = 3.0;
        double r394236 = r394235 + r394221;
        double r394237 = r394224 * r394236;
        double r394238 = r394234 + r394237;
        double r394239 = r394228 * r394238;
        double r394240 = r394227 + r394239;
        double r394241 = r394240 - r394229;
        return r394241;
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))