Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.2

Time: 6.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r369407 = a;
        double r369408 = r369407 * r369407;
        double r369409 = b;
        double r369410 = r369409 * r369409;
        double r369411 = r369408 + r369410;
        double r369412 = 2.0;
        double r369413 = pow(r369411, r369412);
        double r369414 = 4.0;
        double r369415 = 1.0;
        double r369416 = r369415 - r369407;
        double r369417 = r369408 * r369416;
        double r369418 = 3.0;
        double r369419 = r369418 + r369407;
        double r369420 = r369410 * r369419;
        double r369421 = r369417 + r369420;
        double r369422 = r369414 * r369421;
        double r369423 = r369413 + r369422;
        double r369424 = r369423 - r369415;
        return r369424;
}

↓

double f(double a, double b) {
        double r369425 = a;
        double r369426 = r369425 * r369425;
        double r369427 = b;
        double r369428 = r369427 * r369427;
        double r369429 = r369426 + r369428;
        double r369430 = 2.0;
        double r369431 = 2.0;
        double r369432 = r369430 / r369431;
        double r369433 = pow(r369429, r369432);
        double r369434 = 4.0;
        double r369435 = 1.0;
        double r369436 = r369435 - r369425;
        double r369437 = r369426 * r369436;
        double r369438 = 3.0;
        double r369439 = r369438 + r369425;
        double r369440 = r369428 * r369439;
        double r369441 = r369437 + r369440;
        double r369442 = r369434 * r369441;
        double r369443 = fma(r369433, r369433, r369442);
        double r369444 = r369443 - r369435;
        return r369444;
}

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. Using strategy rm

3. Applied sqr-pow 0.2

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

4. Applied fma-def 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))