Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.2

Time: 19.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a, double b) {
        double r151601 = a;
        double r151602 = r151601 * r151601;
        double r151603 = b;
        double r151604 = r151603 * r151603;
        double r151605 = r151602 + r151604;
        double r151606 = 2.0;
        double r151607 = pow(r151605, r151606);
        double r151608 = 4.0;
        double r151609 = 1.0;
        double r151610 = r151609 + r151601;
        double r151611 = r151602 * r151610;
        double r151612 = 3.0;
        double r151613 = r151612 * r151601;
        double r151614 = r151609 - r151613;
        double r151615 = r151604 * r151614;
        double r151616 = r151611 + r151615;
        double r151617 = r151608 * r151616;
        double r151618 = r151607 + r151617;
        double r151619 = r151618 - r151609;
        return r151619;
}

↓

double f(double a, double b) {
        double r151620 = a;
        double r151621 = 2.0;
        double r151622 = pow(r151620, r151621);
        double r151623 = 1.0;
        double r151624 = r151623 + r151620;
        double r151625 = b;
        double r151626 = r151625 * r151625;
        double r151627 = 3.0;
        double r151628 = r151627 * r151620;
        double r151629 = r151623 - r151628;
        double r151630 = r151626 * r151629;
        double r151631 = fma(r151622, r151624, r151630);
        double r151632 = 4.0;
        double r151633 = fma(r151620, r151620, r151626);
        double r151634 = 2.0;
        double r151635 = pow(r151633, r151634);
        double r151636 = fma(r151631, r151632, r151635);
        double r151637 = sqrt(r151636);
        double r151638 = r151637 * r151637;
        double r151639 = r151638 - r151623;
        return r151639;
}

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 add-sqr-sqrt 0.2

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

4. Simplified 0.2

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019351 +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))