Average Error: 0.2 → 0.0
Time: 15.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r2901008 = a;
        double r2901009 = r2901008 * r2901008;
        double r2901010 = b;
        double r2901011 = r2901010 * r2901010;
        double r2901012 = r2901009 + r2901011;
        double r2901013 = 2.0;
        double r2901014 = pow(r2901012, r2901013);
        double r2901015 = 4.0;
        double r2901016 = r2901015 * r2901011;
        double r2901017 = r2901014 + r2901016;
        double r2901018 = 1.0;
        double r2901019 = r2901017 - r2901018;
        return r2901019;
}
↓
double f(double a, double b) {
        double r2901020 = 4.0;
        double r2901021 = b;
        double r2901022 = r2901021 * r2901021;
        double r2901023 = r2901020 * r2901022;
        double r2901024 = cbrt(r2901023);
        double r2901025 = r2901024 * r2901024;
        double r2901026 = r2901024 * r2901025;
        double r2901027 = a;
        double r2901028 = r2901027 * r2901027;
        double r2901029 = r2901022 + r2901028;
        double r2901030 = sqrt(r2901029);
        double r2901031 = pow(r2901030, r2901020);
        double r2901032 = r2901026 + r2901031;
        double r2901033 = 1.0;
        double r2901034 = r2901032 - r2901033;
        return r2901034;
}

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-cube-cbrt0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019154 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))