Average Error: 0.2 → 0.2
Time: 3.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 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
double f(double a, double b) {
        double r314090 = a;
        double r314091 = r314090 * r314090;
        double r314092 = b;
        double r314093 = r314092 * r314092;
        double r314094 = r314091 + r314093;
        double r314095 = 2.0;
        double r314096 = pow(r314094, r314095);
        double r314097 = 4.0;
        double r314098 = r314097 * r314093;
        double r314099 = r314096 + r314098;
        double r314100 = 1.0;
        double r314101 = r314099 - r314100;
        return r314101;
}
↓
double f(double a, double b) {
        double r314102 = a;
        double r314103 = r314102 * r314102;
        double r314104 = b;
        double r314105 = r314104 * r314104;
        double r314106 = r314103 + r314105;
        double r314107 = 2.0;
        double r314108 = pow(r314106, r314107);
        double r314109 = 4.0;
        double r314110 = r314109 * r314105;
        double r314111 = r314108 + r314110;
        double r314112 = 1.0;
        double r314113 = r314111 - r314112;
        return r314113;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))