Average Error: 0.2 → 0.2
Time: 5.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\]
\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
↓
\sqrt{{\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)} \cdot \sqrt{{\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)} - 1
double f(double a, double b) {
        double r249194 = a;
        double r249195 = r249194 * r249194;
        double r249196 = b;
        double r249197 = r249196 * r249196;
        double r249198 = r249195 + r249197;
        double r249199 = 2.0;
        double r249200 = pow(r249198, r249199);
        double r249201 = 4.0;
        double r249202 = 1.0;
        double r249203 = r249202 + r249194;
        double r249204 = r249195 * r249203;
        double r249205 = 3.0;
        double r249206 = r249205 * r249194;
        double r249207 = r249202 - r249206;
        double r249208 = r249197 * r249207;
        double r249209 = r249204 + r249208;
        double r249210 = r249201 * r249209;
        double r249211 = r249200 + r249210;
        double r249212 = r249211 - r249202;
        return r249212;
}
↓
double f(double a, double b) {
        double r249213 = a;
        double r249214 = r249213 * r249213;
        double r249215 = b;
        double r249216 = r249215 * r249215;
        double r249217 = r249214 + r249216;
        double r249218 = 2.0;
        double r249219 = pow(r249217, r249218);
        double r249220 = 4.0;
        double r249221 = 1.0;
        double r249222 = r249221 + r249213;
        double r249223 = r249214 * r249222;
        double r249224 = 3.0;
        double r249225 = r249224 * r249213;
        double r249226 = r249221 - r249225;
        double r249227 = r249216 * r249226;
        double r249228 = r249223 + r249227;
        double r249229 = r249220 * r249228;
        double r249230 = r249219 + r249229;
        double r249231 = sqrt(r249230);
        double r249232 = r249231 * r249231;
        double r249233 = r249232 - r249221;
        return r249233;
}

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 add-sqr-sqrt0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 
(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))