Average Error: 0.2 → 0.2
Time: 3.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
double f(double a, double b) {
        double r202544 = a;
        double r202545 = r202544 * r202544;
        double r202546 = b;
        double r202547 = r202546 * r202546;
        double r202548 = r202545 + r202547;
        double r202549 = 2.0;
        double r202550 = pow(r202548, r202549);
        double r202551 = 4.0;
        double r202552 = r202551 * r202547;
        double r202553 = r202550 + r202552;
        double r202554 = 1.0;
        double r202555 = r202553 - r202554;
        return r202555;
}
↓
double f(double a, double b) {
        double r202556 = a;
        double r202557 = b;
        double r202558 = r202557 * r202557;
        double r202559 = fma(r202556, r202556, r202558);
        double r202560 = 2.0;
        double r202561 = pow(r202559, r202560);
        double r202562 = 4.0;
        double r202563 = r202562 * r202558;
        double r202564 = r202561 + r202563;
        double r202565 = 1.0;
        double r202566 = r202564 - r202565;
        return r202566;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.2

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

  4. Applied pow-unpow0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))