Average Error: 0.2 → 0.2
Time: 6.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r193154 = a;
        double r193155 = r193154 * r193154;
        double r193156 = b;
        double r193157 = r193156 * r193156;
        double r193158 = r193155 + r193157;
        double r193159 = 2.0;
        double r193160 = pow(r193158, r193159);
        double r193161 = 4.0;
        double r193162 = 1.0;
        double r193163 = r193162 + r193154;
        double r193164 = r193155 * r193163;
        double r193165 = 3.0;
        double r193166 = r193165 * r193154;
        double r193167 = r193162 - r193166;
        double r193168 = r193157 * r193167;
        double r193169 = r193164 + r193168;
        double r193170 = r193161 * r193169;
        double r193171 = r193160 + r193170;
        double r193172 = r193171 - r193162;
        return r193172;
}
↓
double f(double a, double b) {
        double r193173 = a;
        double r193174 = r193173 * r193173;
        double r193175 = b;
        double r193176 = r193175 * r193175;
        double r193177 = r193174 + r193176;
        double r193178 = 2.0;
        double r193179 = pow(r193177, r193178);
        double r193180 = 4.0;
        double r193181 = 1.0;
        double r193182 = r193181 * r193181;
        double r193183 = r193182 - r193174;
        double r193184 = r193174 * r193183;
        double r193185 = r193181 - r193173;
        double r193186 = r193184 / r193185;
        double r193187 = 3.0;
        double r193188 = r193187 * r193173;
        double r193189 = r193181 - r193188;
        double r193190 = r193176 * r193189;
        double r193191 = r193186 + r193190;
        double r193192 = r193180 * r193191;
        double r193193 = r193179 + r193192;
        double r193194 = r193193 - r193181;
        return r193194;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip-+0.2

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

  4. Applied associate-*r/0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020003 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))