Average Error: 0.2 → 0.0
Time: 27.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1
double f(double a, double b) {
        double r5087189 = a;
        double r5087190 = r5087189 * r5087189;
        double r5087191 = b;
        double r5087192 = r5087191 * r5087191;
        double r5087193 = r5087190 + r5087192;
        double r5087194 = 2.0;
        double r5087195 = pow(r5087193, r5087194);
        double r5087196 = 4.0;
        double r5087197 = 1.0;
        double r5087198 = r5087197 + r5087189;
        double r5087199 = r5087190 * r5087198;
        double r5087200 = 3.0;
        double r5087201 = r5087200 * r5087189;
        double r5087202 = r5087197 - r5087201;
        double r5087203 = r5087192 * r5087202;
        double r5087204 = r5087199 + r5087203;
        double r5087205 = r5087196 * r5087204;
        double r5087206 = r5087195 + r5087205;
        double r5087207 = r5087206 - r5087197;
        return r5087207;
}
↓
double f(double a, double b) {
        double r5087208 = b;
        double r5087209 = r5087208 * r5087208;
        double r5087210 = a;
        double r5087211 = r5087210 * r5087210;
        double r5087212 = r5087209 + r5087211;
        double r5087213 = sqrt(r5087212);
        double r5087214 = 4.0;
        double r5087215 = pow(r5087213, r5087214);
        double r5087216 = -3.0;
        double r5087217 = r5087216 * r5087209;
        double r5087218 = r5087210 + r5087211;
        double r5087219 = r5087217 + r5087218;
        double r5087220 = r5087219 * r5087210;
        double r5087221 = r5087220 + r5087209;
        double r5087222 = r5087221 * r5087214;
        double r5087223 = r5087215 + r5087222;
        double r5087224 = -1.0;
        double r5087225 = r5087223 + r5087224;
        return r5087225;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  9. Applied pow-plus0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 
(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))