Average Error: 0.2 → 0.2
Time: 22.5s
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 r231484 = a;
        double r231485 = r231484 * r231484;
        double r231486 = b;
        double r231487 = r231486 * r231486;
        double r231488 = r231485 + r231487;
        double r231489 = 2.0;
        double r231490 = pow(r231488, r231489);
        double r231491 = 4.0;
        double r231492 = 1.0;
        double r231493 = r231492 + r231484;
        double r231494 = r231485 * r231493;
        double r231495 = 3.0;
        double r231496 = r231495 * r231484;
        double r231497 = r231492 - r231496;
        double r231498 = r231487 * r231497;
        double r231499 = r231494 + r231498;
        double r231500 = r231491 * r231499;
        double r231501 = r231490 + r231500;
        double r231502 = r231501 - r231492;
        return r231502;
}
↓
double f(double a, double b) {
        double r231503 = 4.0;
        double r231504 = a;
        double r231505 = r231504 * r231504;
        double r231506 = 1.0;
        double r231507 = r231506 + r231504;
        double r231508 = b;
        double r231509 = r231508 * r231508;
        double r231510 = 3.0;
        double r231511 = r231510 * r231504;
        double r231512 = r231506 - r231511;
        double r231513 = r231509 * r231512;
        double r231514 = fma(r231505, r231507, r231513);
        double r231515 = fma(r231504, r231504, r231509);
        double r231516 = 2.0;
        double r231517 = pow(r231515, r231516);
        double r231518 = fma(r231503, r231514, r231517);
        double r231519 = r231518 - r231506;
        return r231519;
}

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 2019208 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))