Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 5.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r246167 = a;
        double r246168 = r246167 * r246167;
        double r246169 = b;
        double r246170 = r246169 * r246169;
        double r246171 = r246168 + r246170;
        double r246172 = 2.0;
        double r246173 = pow(r246171, r246172);
        double r246174 = 4.0;
        double r246175 = r246174 * r246170;
        double r246176 = r246173 + r246175;
        double r246177 = 1.0;
        double r246178 = r246176 - r246177;
        return r246178;
}

↓

double f(double a, double b) {
        double r246179 = 4.0;
        double r246180 = b;
        double r246181 = r246179 * r246180;
        double r246182 = a;
        double r246183 = hypot(r246182, r246180);
        double r246184 = 2.0;
        double r246185 = 2.0;
        double r246186 = r246184 * r246185;
        double r246187 = pow(r246183, r246186);
        double r246188 = 1.0;
        double r246189 = r246187 - r246188;
        double r246190 = -r246188;
        double r246191 = 1.0;
        double r246192 = fma(r246190, r246191, r246188);
        double r246193 = r246189 + r246192;
        double r246194 = fma(r246181, r246180, r246193);
        return r246194;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

4. Applied add-cube-cbrt 0.2

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

5. Applied add-sqr-sqrt 0.2

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

6. Applied unpow-prod-down 0.2

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

7. Applied prod-diff 0.2

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

8. Simplified 0.0

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

9. Simplified 0.0

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

10. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))