Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 6.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r180076 = a;
        double r180077 = r180076 * r180076;
        double r180078 = b;
        double r180079 = r180078 * r180078;
        double r180080 = r180077 + r180079;
        double r180081 = 2.0;
        double r180082 = pow(r180080, r180081);
        double r180083 = 4.0;
        double r180084 = 1.0;
        double r180085 = r180084 - r180076;
        double r180086 = r180077 * r180085;
        double r180087 = 3.0;
        double r180088 = r180087 + r180076;
        double r180089 = r180079 * r180088;
        double r180090 = r180086 + r180089;
        double r180091 = r180083 * r180090;
        double r180092 = r180082 + r180091;
        double r180093 = r180092 - r180084;
        return r180093;
}

↓

double f(double a, double b) {
        double r180094 = a;
        double r180095 = b;
        double r180096 = r180095 * r180095;
        double r180097 = fma(r180094, r180094, r180096);
        double r180098 = 2.0;
        double r180099 = 2.0;
        double r180100 = r180098 / r180099;
        double r180101 = pow(r180097, r180100);
        double r180102 = sqrt(r180101);
        double r180103 = 4.0;
        double r180104 = pow(r180102, r180103);
        double r180105 = 4.0;
        double r180106 = r180094 * r180094;
        double r180107 = 1.0;
        double r180108 = r180107 - r180094;
        double r180109 = r180106 * r180108;
        double r180110 = 3.0;
        double r180111 = r180110 + r180094;
        double r180112 = r180096 * r180111;
        double r180113 = r180109 + r180112;
        double r180114 = r180105 * r180113;
        double r180115 = r180104 + r180114;
        double r180116 = r180115 - r180107;
        return r180116;
}

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. Simplified 0.2

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

5. Simplified 0.2

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

7. Applied add-sqr-sqrt 0.2

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

8. Applied associate-*l* 0.1

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

9. Simplified 0.1

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

11. Applied pow1 0.1

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

12. Applied pow-prod-up 0.0

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

13. Simplified 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020046 +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))