Average Error: 0.2 → 0.0
Time: 44.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(e^{\log \left(b \cdot b\right) + \log 4} + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(e^{\log \left(b \cdot b\right) + \log 4} + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r51495193 = a;
        double r51495194 = r51495193 * r51495193;
        double r51495195 = b;
        double r51495196 = r51495195 * r51495195;
        double r51495197 = r51495194 + r51495196;
        double r51495198 = 2.0;
        double r51495199 = pow(r51495197, r51495198);
        double r51495200 = 4.0;
        double r51495201 = r51495200 * r51495196;
        double r51495202 = r51495199 + r51495201;
        double r51495203 = 1.0;
        double r51495204 = r51495202 - r51495203;
        return r51495204;
}
↓
double f(double a, double b) {
        double r51495205 = b;
        double r51495206 = r51495205 * r51495205;
        double r51495207 = log(r51495206);
        double r51495208 = 4.0;
        double r51495209 = log(r51495208);
        double r51495210 = r51495207 + r51495209;
        double r51495211 = exp(r51495210);
        double r51495212 = a;
        double r51495213 = r51495212 * r51495212;
        double r51495214 = r51495206 + r51495213;
        double r51495215 = sqrt(r51495214);
        double r51495216 = pow(r51495215, r51495208);
        double r51495217 = r51495211 + r51495216;
        double r51495218 = 1.0;
        double r51495219 = r51495217 - r51495218;
        return r51495219;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  13. Applied add-exp-log0.0

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

  14. Applied add-exp-log0.0

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

  15. Applied prod-exp0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019107 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))