Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 22.2s

Precision: 64

Math

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

↓

\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\right) - 1\]

C

double f(double a, double b) {
        double r7051522 = a;
        double r7051523 = r7051522 * r7051522;
        double r7051524 = b;
        double r7051525 = r7051524 * r7051524;
        double r7051526 = r7051523 + r7051525;
        double r7051527 = 2.0;
        double r7051528 = pow(r7051526, r7051527);
        double r7051529 = 4.0;
        double r7051530 = 1.0;
        double r7051531 = r7051530 + r7051522;
        double r7051532 = r7051523 * r7051531;
        double r7051533 = 3.0;
        double r7051534 = r7051533 * r7051522;
        double r7051535 = r7051530 - r7051534;
        double r7051536 = r7051525 * r7051535;
        double r7051537 = r7051532 + r7051536;
        double r7051538 = r7051529 * r7051537;
        double r7051539 = r7051528 + r7051538;
        double r7051540 = r7051539 - r7051530;
        return r7051540;
}

↓

double f(double a, double b) {
        double r7051541 = a;
        double r7051542 = r7051541 * r7051541;
        double r7051543 = r7051542 * r7051541;
        double r7051544 = b;
        double r7051545 = r7051544 * r7051544;
        double r7051546 = r7051545 + r7051542;
        double r7051547 = r7051543 + r7051546;
        double r7051548 = 4.0;
        double r7051549 = r7051547 * r7051548;
        double r7051550 = -12.0;
        double r7051551 = r7051545 * r7051541;
        double r7051552 = r7051550 * r7051551;
        double r7051553 = sqrt(r7051546);
        double r7051554 = pow(r7051553, r7051548);
        double r7051555 = r7051552 + r7051554;
        double r7051556 = r7051549 + r7051555;
        double r7051557 = 1.0;
        double r7051558 = r7051556 - r7051557;
        return r7051558;
}

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. Simplified 0.2

   \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

5. Applied associate-*r* 0.1

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]
6. Using strategy rm

7. Applied pow1 0.1

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

8. Applied add-sqr-sqrt 0.1

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

9. Applied pow3 0.1

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{1} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

10. Applied pow-prod-up 0.0

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\left(3 + 1\right)}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

11. Simplified 0.0

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{\color{blue}{4}} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

12. Final simplification 0.0

    \[\leadsto \left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(-12 \cdot \left(\left(b \cdot b\right) \cdot a\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right)\right) - 1\]

Reproduce

herbie shell --seed 2019152 
(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))