Average Error: 0.2 → 0.2
Time: 6.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(\left(a \cdot a\right) \cdot \left(\sqrt[3]{1 \cdot 1 - a \cdot a} \cdot \sqrt[3]{1 \cdot 1 - a \cdot a}\right)\right) \cdot \sqrt[3]{1 \cdot 1 - a \cdot a}}{1 + a} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(\left(a \cdot a\right) \cdot \left(\sqrt[3]{1 \cdot 1 - a \cdot a} \cdot \sqrt[3]{1 \cdot 1 - a \cdot a}\right)\right) \cdot \sqrt[3]{1 \cdot 1 - a \cdot a}}{1 + a} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
double f(double a, double b) {
        double r275354 = a;
        double r275355 = r275354 * r275354;
        double r275356 = b;
        double r275357 = r275356 * r275356;
        double r275358 = r275355 + r275357;
        double r275359 = 2.0;
        double r275360 = pow(r275358, r275359);
        double r275361 = 4.0;
        double r275362 = 1.0;
        double r275363 = r275362 - r275354;
        double r275364 = r275355 * r275363;
        double r275365 = 3.0;
        double r275366 = r275365 + r275354;
        double r275367 = r275357 * r275366;
        double r275368 = r275364 + r275367;
        double r275369 = r275361 * r275368;
        double r275370 = r275360 + r275369;
        double r275371 = r275370 - r275362;
        return r275371;
}
↓
double f(double a, double b) {
        double r275372 = a;
        double r275373 = r275372 * r275372;
        double r275374 = b;
        double r275375 = r275374 * r275374;
        double r275376 = r275373 + r275375;
        double r275377 = 2.0;
        double r275378 = pow(r275376, r275377);
        double r275379 = 4.0;
        double r275380 = 1.0;
        double r275381 = r275380 * r275380;
        double r275382 = r275381 - r275373;
        double r275383 = cbrt(r275382);
        double r275384 = r275383 * r275383;
        double r275385 = r275373 * r275384;
        double r275386 = r275385 * r275383;
        double r275387 = r275380 + r275372;
        double r275388 = r275386 / r275387;
        double r275389 = 3.0;
        double r275390 = r275389 + r275372;
        double r275391 = r275375 * r275390;
        double r275392 = r275388 + r275391;
        double r275393 = r275379 * r275392;
        double r275394 = r275378 + r275393;
        double r275395 = r275394 - r275380;
        return r275395;
}

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

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

  3. Applied flip--0.2

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

  4. Applied associate-*r/0.2

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

  6. Applied add-cube-cbrt0.2

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

  7. Applied associate-*r*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019356 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))