Average Error: 0.2 → 0.1
Time: 17.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3} \cdot \sqrt{a \cdot a + b \cdot b} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right)\right) - 1\]
\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(\sqrt{a \cdot a + b \cdot b}\right)}^{3} \cdot \sqrt{a \cdot a + b \cdot b} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(1 - a\right) \cdot \left(a \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r8870195 = a;
        double r8870196 = r8870195 * r8870195;
        double r8870197 = b;
        double r8870198 = r8870197 * r8870197;
        double r8870199 = r8870196 + r8870198;
        double r8870200 = 2.0;
        double r8870201 = pow(r8870199, r8870200);
        double r8870202 = 4.0;
        double r8870203 = 1.0;
        double r8870204 = r8870203 - r8870195;
        double r8870205 = r8870196 * r8870204;
        double r8870206 = 3.0;
        double r8870207 = r8870206 + r8870195;
        double r8870208 = r8870198 * r8870207;
        double r8870209 = r8870205 + r8870208;
        double r8870210 = r8870202 * r8870209;
        double r8870211 = r8870201 + r8870210;
        double r8870212 = r8870211 - r8870203;
        return r8870212;
}
↓
double f(double a, double b) {
        double r8870213 = a;
        double r8870214 = r8870213 * r8870213;
        double r8870215 = b;
        double r8870216 = r8870215 * r8870215;
        double r8870217 = r8870214 + r8870216;
        double r8870218 = sqrt(r8870217);
        double r8870219 = 3.0;
        double r8870220 = pow(r8870218, r8870219);
        double r8870221 = r8870220 * r8870218;
        double r8870222 = 4.0;
        double r8870223 = r8870213 + r8870219;
        double r8870224 = r8870216 * r8870223;
        double r8870225 = 1.0;
        double r8870226 = r8870225 - r8870213;
        double r8870227 = r8870226 * r8870214;
        double r8870228 = r8870224 + r8870227;
        double r8870229 = r8870222 * r8870228;
        double r8870230 = r8870221 + r8870229;
        double r8870231 = r8870230 - r8870225;
        return r8870231;
}

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 unpow20.2

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

  4. Applied fma-def0.2

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

  6. Applied fma-udef0.2

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

  8. Applied add-sqr-sqrt0.2

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

  9. Applied associate-*l*0.1

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

  11. Applied add-sqr-sqrt0.1

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

  12. Applied cube-unmult0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(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))