Average Error: 0.2 → 0.0
Time: 21.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(\left({a}^{4} + e^{\log \left(\left(a \cdot b\right) \cdot \left(2 \cdot \left(a \cdot b\right)\right)\right)}\right) + {b}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left(\left({a}^{4} + e^{\log \left(\left(a \cdot b\right) \cdot \left(2 \cdot \left(a \cdot b\right)\right)\right)}\right) + {b}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1
double f(double a, double b) {
        double r8130276 = a;
        double r8130277 = r8130276 * r8130276;
        double r8130278 = b;
        double r8130279 = r8130278 * r8130278;
        double r8130280 = r8130277 + r8130279;
        double r8130281 = 2.0;
        double r8130282 = pow(r8130280, r8130281);
        double r8130283 = 4.0;
        double r8130284 = r8130283 * r8130279;
        double r8130285 = r8130282 + r8130284;
        double r8130286 = 1.0;
        double r8130287 = r8130285 - r8130286;
        return r8130287;
}
↓
double f(double a, double b) {
        double r8130288 = a;
        double r8130289 = 4.0;
        double r8130290 = pow(r8130288, r8130289);
        double r8130291 = b;
        double r8130292 = r8130288 * r8130291;
        double r8130293 = 2.0;
        double r8130294 = r8130293 * r8130292;
        double r8130295 = r8130292 * r8130294;
        double r8130296 = log(r8130295);
        double r8130297 = exp(r8130296);
        double r8130298 = r8130290 + r8130297;
        double r8130299 = pow(r8130291, r8130289);
        double r8130300 = r8130298 + r8130299;
        double r8130301 = r8130291 * r8130291;
        double r8130302 = r8130289 * r8130301;
        double r8130303 = r8130300 + r8130302;
        double r8130304 = 1.0;
        double r8130305 = r8130303 - r8130304;
        return r8130305;
}

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. Taylor expanded around 0 0.0

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

  4. Simplified0.0

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \color{blue}{\left(\left({a}^{4} + \left(2 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot a\right)\right) + {b}^{4}\right)}\right) - 1\]
  5. Using strategy rm

  6. Applied add-exp-log22.2

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

  7. Applied add-exp-log22.3

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

  8. Applied add-exp-log22.3

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

  9. Applied prod-exp22.3

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

  10. Applied prod-exp22.3

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))