Average Error: 0.2 → 0.7
Time: 13.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{a \cdot a + b \cdot b}}\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\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(\sqrt[3]{a \cdot a + b \cdot b} \cdot \sqrt[3]{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{a \cdot a + b \cdot b}} \cdot \sqrt[3]{\sqrt{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
double f(double a, double b) {
        double r362441 = a;
        double r362442 = r362441 * r362441;
        double r362443 = b;
        double r362444 = r362443 * r362443;
        double r362445 = r362442 + r362444;
        double r362446 = 2.0;
        double r362447 = pow(r362445, r362446);
        double r362448 = 4.0;
        double r362449 = 1.0;
        double r362450 = r362449 + r362441;
        double r362451 = r362442 * r362450;
        double r362452 = 3.0;
        double r362453 = r362452 * r362441;
        double r362454 = r362449 - r362453;
        double r362455 = r362444 * r362454;
        double r362456 = r362451 + r362455;
        double r362457 = r362448 * r362456;
        double r362458 = r362447 + r362457;
        double r362459 = r362458 - r362449;
        return r362459;
}
↓
double f(double a, double b) {
        double r362460 = a;
        double r362461 = r362460 * r362460;
        double r362462 = b;
        double r362463 = r362462 * r362462;
        double r362464 = r362461 + r362463;
        double r362465 = cbrt(r362464);
        double r362466 = r362465 * r362465;
        double r362467 = 2.0;
        double r362468 = pow(r362466, r362467);
        double r362469 = sqrt(r362464);
        double r362470 = cbrt(r362469);
        double r362471 = r362470 * r362470;
        double r362472 = pow(r362471, r362467);
        double r362473 = r362468 * r362472;
        double r362474 = 4.0;
        double r362475 = 1.0;
        double r362476 = r362475 + r362460;
        double r362477 = r362461 * r362476;
        double r362478 = 3.0;
        double r362479 = r362478 * r362460;
        double r362480 = r362475 - r362479;
        double r362481 = r362463 * r362480;
        double r362482 = r362477 + r362481;
        double r362483 = r362474 * r362482;
        double r362484 = r362473 + r362483;
        double r362485 = r362484 - r362475;
        return r362485;
}

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-cube-cbrt0.7

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

  4. Applied unpow-prod-down0.7

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

  6. Applied add-sqr-sqrt0.7

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

  7. Applied cbrt-prod0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 
(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))