Average Error: 0.2 → 0.1
Time: 40.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left((b \cdot \left((b \cdot b + \left(\left(a \cdot a\right) \cdot 2\right))_* \cdot b\right) + \left({a}^{4}\right))_* + \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4\right) - 1\]
double f(double a, double b) {
        double r55241093 = a;
        double r55241094 = r55241093 * r55241093;
        double r55241095 = b;
        double r55241096 = r55241095 * r55241095;
        double r55241097 = r55241094 + r55241096;
        double r55241098 = 2.0;
        double r55241099 = pow(r55241097, r55241098);
        double r55241100 = 4.0;
        double r55241101 = 1.0;
        double r55241102 = r55241101 - r55241093;
        double r55241103 = r55241094 * r55241102;
        double r55241104 = 3.0;
        double r55241105 = r55241104 + r55241093;
        double r55241106 = r55241096 * r55241105;
        double r55241107 = r55241103 + r55241106;
        double r55241108 = r55241100 * r55241107;
        double r55241109 = r55241099 + r55241108;
        double r55241110 = r55241109 - r55241101;
        return r55241110;
}
↓
double f(double a, double b) {
        double r55241111 = b;
        double r55241112 = a;
        double r55241113 = r55241112 * r55241112;
        double r55241114 = 2.0;
        double r55241115 = r55241113 * r55241114;
        double r55241116 = fma(r55241111, r55241111, r55241115);
        double r55241117 = r55241116 * r55241111;
        double r55241118 = 4.0;
        double r55241119 = pow(r55241112, r55241118);
        double r55241120 = fma(r55241111, r55241117, r55241119);
        double r55241121 = r55241111 * r55241111;
        double r55241122 = 3.0;
        double r55241123 = r55241122 + r55241112;
        double r55241124 = r55241121 * r55241123;
        double r55241125 = 1.0;
        double r55241126 = r55241125 - r55241112;
        double r55241127 = r55241113 * r55241126;
        double r55241128 = r55241124 + r55241127;
        double r55241129 = r55241128 * r55241118;
        double r55241130 = r55241120 + r55241129;
        double r55241131 = r55241130 - r55241125;
        return r55241131;
}


\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
↓
\left((b \cdot \left((b \cdot b + \left(\left(a \cdot a\right) \cdot 2\right))_* \cdot b\right) + \left({a}^{4}\right))_* + \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4\right) - 1

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  3. Simplified0.2

    \[\leadsto \left(\color{blue}{(b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right))_*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  4. Taylor expanded around -inf 0.1

    \[\leadsto \left((b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \color{blue}{\left({a}^{4}\right)})_* + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  5. Final simplification0.1

    \[\leadsto \left((b \cdot \left((b \cdot b + \left(\left(a \cdot a\right) \cdot 2\right))_* \cdot b\right) + \left({a}^{4}\right))_* + \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4\right) - 1\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))