Average Error: 0.2 → 0.0
Time: 25.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + e^{\log \left(4 \cdot \left(b \cdot b\right)\right)}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + e^{\log \left(4 \cdot \left(b \cdot b\right)\right)}\right) - 1
double f(double a, double b) {
        double r7572025 = a;
        double r7572026 = r7572025 * r7572025;
        double r7572027 = b;
        double r7572028 = r7572027 * r7572027;
        double r7572029 = r7572026 + r7572028;
        double r7572030 = 2.0;
        double r7572031 = pow(r7572029, r7572030);
        double r7572032 = 4.0;
        double r7572033 = r7572032 * r7572028;
        double r7572034 = r7572031 + r7572033;
        double r7572035 = 1.0;
        double r7572036 = r7572034 - r7572035;
        return r7572036;
}
↓
double f(double a, double b) {
        double r7572037 = a;
        double r7572038 = r7572037 * r7572037;
        double r7572039 = b;
        double r7572040 = r7572039 * r7572039;
        double r7572041 = r7572038 + r7572040;
        double r7572042 = sqrt(r7572041);
        double r7572043 = 4.0;
        double r7572044 = pow(r7572042, r7572043);
        double r7572045 = r7572043 * r7572040;
        double r7572046 = log(r7572045);
        double r7572047 = exp(r7572046);
        double r7572048 = r7572044 + r7572047;
        double r7572049 = 1.0;
        double r7572050 = r7572048 - r7572049;
        return r7572050;
}

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

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))