Average Error: 0.2 → 0.2
Time: 29.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r10932068 = a;
        double r10932069 = r10932068 * r10932068;
        double r10932070 = b;
        double r10932071 = r10932070 * r10932070;
        double r10932072 = r10932069 + r10932071;
        double r10932073 = 2.0;
        double r10932074 = pow(r10932072, r10932073);
        double r10932075 = 4.0;
        double r10932076 = 1.0;
        double r10932077 = r10932076 + r10932068;
        double r10932078 = r10932069 * r10932077;
        double r10932079 = 3.0;
        double r10932080 = r10932079 * r10932068;
        double r10932081 = r10932076 - r10932080;
        double r10932082 = r10932071 * r10932081;
        double r10932083 = r10932078 + r10932082;
        double r10932084 = r10932075 * r10932083;
        double r10932085 = r10932074 + r10932084;
        double r10932086 = r10932085 - r10932076;
        return r10932086;
}
↓
double f(double a, double b) {
        double r10932087 = 4.0;
        double r10932088 = a;
        double r10932089 = r10932088 * r10932088;
        double r10932090 = 1.0;
        double r10932091 = r10932090 + r10932088;
        double r10932092 = b;
        double r10932093 = r10932092 * r10932092;
        double r10932094 = 3.0;
        double r10932095 = r10932094 * r10932088;
        double r10932096 = r10932090 - r10932095;
        double r10932097 = r10932093 * r10932096;
        double r10932098 = fma(r10932089, r10932091, r10932097);
        double r10932099 = fma(r10932088, r10932088, r10932093);
        double r10932100 = 2.0;
        double r10932101 = pow(r10932099, r10932100);
        double r10932102 = fma(r10932087, r10932098, r10932101);
        double r10932103 = r10932102 - r10932090;
        return r10932103;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))