Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.5

Time: 25.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r9489058 = a;
        double r9489059 = r9489058 * r9489058;
        double r9489060 = b;
        double r9489061 = r9489060 * r9489060;
        double r9489062 = r9489059 + r9489061;
        double r9489063 = 2.0;
        double r9489064 = pow(r9489062, r9489063);
        double r9489065 = 4.0;
        double r9489066 = r9489065 * r9489061;
        double r9489067 = r9489064 + r9489066;
        double r9489068 = 1.0;
        double r9489069 = r9489067 - r9489068;
        return r9489069;
}

↓

double f(double a, double b) {
        double r9489070 = b;
        double r9489071 = r9489070 * r9489070;
        double r9489072 = 4.0;
        double r9489073 = r9489071 * r9489072;
        double r9489074 = a;
        double r9489075 = r9489074 * r9489074;
        double r9489076 = r9489075 + r9489071;
        double r9489077 = 2.0;
        double r9489078 = pow(r9489076, r9489077);
        double r9489079 = r9489073 + r9489078;
        double r9489080 = sqrt(r9489079);
        double r9489081 = cbrt(r9489080);
        double r9489082 = cbrt(r9489079);
        double r9489083 = r9489081 * r9489081;
        double r9489084 = r9489083 * r9489081;
        double r9489085 = cbrt(r9489084);
        double r9489086 = r9489082 * r9489085;
        double r9489087 = r9489081 * r9489086;
        double r9489088 = sqrt(r9489082);
        double r9489089 = r9489088 * r9489088;
        double r9489090 = r9489087 * r9489089;
        double r9489091 = 1.0;
        double r9489092 = r9489090 - r9489091;
        return r9489092;
}

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. Using strategy rm

3. Applied add-cube-cbrt 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied cbrt-prod 0.5

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

7. Applied associate-*r* 0.5

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

9. Applied add-cube-cbrt 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019171 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))