Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 52.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r14742649 = a;
        double r14742650 = r14742649 * r14742649;
        double r14742651 = b;
        double r14742652 = r14742651 * r14742651;
        double r14742653 = r14742650 + r14742652;
        double r14742654 = 2.0;
        double r14742655 = pow(r14742653, r14742654);
        double r14742656 = 4.0;
        double r14742657 = 1.0;
        double r14742658 = r14742657 + r14742649;
        double r14742659 = r14742650 * r14742658;
        double r14742660 = 3.0;
        double r14742661 = r14742660 * r14742649;
        double r14742662 = r14742657 - r14742661;
        double r14742663 = r14742652 * r14742662;
        double r14742664 = r14742659 + r14742663;
        double r14742665 = r14742656 * r14742664;
        double r14742666 = r14742655 + r14742665;
        double r14742667 = r14742666 - r14742657;
        return r14742667;
}

↓

double f(double a, double b) {
        double r14742668 = 4.0;
        double r14742669 = a;
        double r14742670 = r14742669 * r14742669;
        double r14742671 = 1.0;
        double r14742672 = r14742671 + r14742669;
        double r14742673 = r14742670 * r14742672;
        double r14742674 = b;
        double r14742675 = r14742674 * r14742674;
        double r14742676 = 3.0;
        double r14742677 = r14742669 * r14742676;
        double r14742678 = r14742671 - r14742677;
        double r14742679 = r14742675 * r14742678;
        double r14742680 = r14742673 + r14742679;
        double r14742681 = r14742668 * r14742680;
        double r14742682 = 2.0;
        double r14742683 = r14742670 * r14742682;
        double r14742684 = fma(r14742674, r14742674, r14742683);
        double r14742685 = r14742684 * r14742674;
        double r14742686 = pow(r14742669, r14742668);
        double r14742687 = fma(r14742674, r14742685, r14742686);
        double r14742688 = r14742681 + r14742687;
        double r14742689 = r14742688 - r14742671;
        return r14742689;
}

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. Taylor expanded around 0 0.0

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

3. Simplified 0.2

   \[\leadsto \left(\color{blue}{(b \cdot \left(b \cdot (b \cdot b + \left(2 \cdot \left(a \cdot a\right)\right))_*\right) + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right))_*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
4. Using strategy rm

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-sqr 0.2

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

8. Applied pow1 0.2

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

9. Applied pow-plus 0.2

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

10. Applied pow-prod-up 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019104 +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))