Average Error: 0.2 → 0.2
Time: 46.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \left(4 \cdot a\right) \cdot \mathsf{fma}\left(a, a, a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) - 1\]
\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
↓
\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), \left(4 \cdot a\right) \cdot \mathsf{fma}\left(a, a, a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) - 1
double f(double a, double b) {
        double r5345533 = a;
        double r5345534 = r5345533 * r5345533;
        double r5345535 = b;
        double r5345536 = r5345535 * r5345535;
        double r5345537 = r5345534 + r5345536;
        double r5345538 = 2.0;
        double r5345539 = pow(r5345537, r5345538);
        double r5345540 = 4.0;
        double r5345541 = 1.0;
        double r5345542 = r5345541 + r5345533;
        double r5345543 = r5345534 * r5345542;
        double r5345544 = 3.0;
        double r5345545 = r5345544 * r5345533;
        double r5345546 = r5345541 - r5345545;
        double r5345547 = r5345536 * r5345546;
        double r5345548 = r5345543 + r5345547;
        double r5345549 = r5345540 * r5345548;
        double r5345550 = r5345539 + r5345549;
        double r5345551 = r5345550 - r5345541;
        return r5345551;
}
↓
double f(double a, double b) {
        double r5345552 = a;
        double r5345553 = b;
        double r5345554 = r5345553 * r5345553;
        double r5345555 = fma(r5345552, r5345552, r5345554);
        double r5345556 = 4.0;
        double r5345557 = r5345556 * r5345552;
        double r5345558 = fma(r5345552, r5345552, r5345552);
        double r5345559 = r5345557 * r5345558;
        double r5345560 = fma(r5345555, r5345555, r5345559);
        double r5345561 = 1.0;
        double r5345562 = 3.0;
        double r5345563 = r5345552 * r5345562;
        double r5345564 = r5345561 - r5345563;
        double r5345565 = r5345554 * r5345564;
        double r5345566 = r5345556 * r5345565;
        double r5345567 = r5345560 + r5345566;
        double r5345568 = r5345567 - r5345561;
        return r5345568;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))