Average Error: 0.2 → 0.2
Time: 16.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}} \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}}\right) \cdot \sqrt{4 \cdot \left(\left(1 - a \cdot 3\right) \cdot {b}^{2} + a \cdot \left(\left(a + 1\right) \cdot a\right)\right) + {\left({b}^{2} + a \cdot a\right)}^{2}} - 1\]
\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(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}} \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \left(\left(1 - a \cdot 3\right) \cdot b\right) + a \cdot \left(\left(a + 1\right) \cdot a\right)\right)}}\right) \cdot \sqrt{4 \cdot \left(\left(1 - a \cdot 3\right) \cdot {b}^{2} + a \cdot \left(\left(a + 1\right) \cdot a\right)\right) + {\left({b}^{2} + a \cdot a\right)}^{2}} - 1
double f(double a, double b) {
        double r240259 = a;
        double r240260 = r240259 * r240259;
        double r240261 = b;
        double r240262 = r240261 * r240261;
        double r240263 = r240260 + r240262;
        double r240264 = 2.0;
        double r240265 = pow(r240263, r240264);
        double r240266 = 4.0;
        double r240267 = 1.0;
        double r240268 = r240267 + r240259;
        double r240269 = r240260 * r240268;
        double r240270 = 3.0;
        double r240271 = r240270 * r240259;
        double r240272 = r240267 - r240271;
        double r240273 = r240262 * r240272;
        double r240274 = r240269 + r240273;
        double r240275 = r240266 * r240274;
        double r240276 = r240265 + r240275;
        double r240277 = r240276 - r240267;
        return r240277;
}
↓
double f(double a, double b) {
        double r240278 = a;
        double r240279 = r240278 * r240278;
        double r240280 = b;
        double r240281 = r240280 * r240280;
        double r240282 = r240279 + r240281;
        double r240283 = 2.0;
        double r240284 = pow(r240282, r240283);
        double r240285 = 4.0;
        double r240286 = 1.0;
        double r240287 = 3.0;
        double r240288 = r240278 * r240287;
        double r240289 = r240286 - r240288;
        double r240290 = r240289 * r240280;
        double r240291 = r240280 * r240290;
        double r240292 = r240278 + r240286;
        double r240293 = r240292 * r240278;
        double r240294 = r240278 * r240293;
        double r240295 = r240291 + r240294;
        double r240296 = r240285 * r240295;
        double r240297 = r240284 + r240296;
        double r240298 = sqrt(r240297);
        double r240299 = sqrt(r240298);
        double r240300 = r240299 * r240299;
        double r240301 = 2.0;
        double r240302 = pow(r240280, r240301);
        double r240303 = r240289 * r240302;
        double r240304 = r240303 + r240294;
        double r240305 = r240285 * r240304;
        double r240306 = r240302 + r240279;
        double r240307 = pow(r240306, r240283);
        double r240308 = r240305 + r240307;
        double r240309 = sqrt(r240308);
        double r240310 = r240300 * r240309;
        double r240311 = r240310 - r240286;
        return r240311;
}

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. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.2

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019196 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))