Average Error: 0.2 → 0.0
Time: 18.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4\]
\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(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4
double f(double a, double b) {
        double r6131326 = a;
        double r6131327 = r6131326 * r6131326;
        double r6131328 = b;
        double r6131329 = r6131328 * r6131328;
        double r6131330 = r6131327 + r6131329;
        double r6131331 = 2.0;
        double r6131332 = pow(r6131330, r6131331);
        double r6131333 = 4.0;
        double r6131334 = 1.0;
        double r6131335 = r6131334 + r6131326;
        double r6131336 = r6131327 * r6131335;
        double r6131337 = 3.0;
        double r6131338 = r6131337 * r6131326;
        double r6131339 = r6131334 - r6131338;
        double r6131340 = r6131329 * r6131339;
        double r6131341 = r6131336 + r6131340;
        double r6131342 = r6131333 * r6131341;
        double r6131343 = r6131332 + r6131342;
        double r6131344 = r6131343 - r6131334;
        return r6131344;
}
↓
double f(double a, double b) {
        double r6131345 = 4.0;
        double r6131346 = a;
        double r6131347 = r6131346 * r6131346;
        double r6131348 = b;
        double r6131349 = r6131348 * r6131348;
        double r6131350 = r6131347 + r6131349;
        double r6131351 = -3.0;
        double r6131352 = r6131346 * r6131349;
        double r6131353 = r6131351 * r6131352;
        double r6131354 = r6131350 + r6131353;
        double r6131355 = r6131345 * r6131354;
        double r6131356 = -1.0;
        double r6131357 = sqrt(r6131350);
        double r6131358 = pow(r6131357, r6131345);
        double r6131359 = r6131356 + r6131358;
        double r6131360 = r6131355 + r6131359;
        double r6131361 = cbrt(r6131346);
        double r6131362 = r6131361 * r6131361;
        double r6131363 = r6131362 * r6131347;
        double r6131364 = r6131361 * r6131363;
        double r6131365 = r6131364 * r6131345;
        double r6131366 = r6131360 + r6131365;
        return r6131366;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  14. Applied add-cube-cbrt0.0

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

  15. Applied associate-*r*0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))