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 r308362 = a;
        double r308363 = r308362 * r308362;
        double r308364 = b;
        double r308365 = r308364 * r308364;
        double r308366 = r308363 + r308365;
        double r308367 = 2.0;
        double r308368 = pow(r308366, r308367);
        double r308369 = 4.0;
        double r308370 = 1.0;
        double r308371 = r308370 + r308362;
        double r308372 = r308363 * r308371;
        double r308373 = 3.0;
        double r308374 = r308373 * r308362;
        double r308375 = r308370 - r308374;
        double r308376 = r308365 * r308375;
        double r308377 = r308372 + r308376;
        double r308378 = r308369 * r308377;
        double r308379 = r308368 + r308378;
        double r308380 = r308379 - r308370;
        return r308380;
}
↓
double f(double a, double b) {
        double r308381 = 4.0;
        double r308382 = a;
        double r308383 = r308382 * r308382;
        double r308384 = 1.0;
        double r308385 = r308384 + r308382;
        double r308386 = b;
        double r308387 = r308386 * r308386;
        double r308388 = 3.0;
        double r308389 = r308388 * r308382;
        double r308390 = r308384 - r308389;
        double r308391 = r308387 * r308390;
        double r308392 = fma(r308383, r308385, r308391);
        double r308393 = fma(r308382, r308382, r308387);
        double r308394 = 2.0;
        double r308395 = pow(r308393, r308394);
        double r308396 = fma(r308381, r308392, r308395);
        double r308397 = r308396 - r308384;
        return r308397;
}

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 2019325 +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))