Average Error: 0.2 → 0.1
Time: 20.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(b \cdot b\right) \cdot 12 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3} \cdot \sqrt{a \cdot a + b \cdot b}\right) + 4 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right)\right) + -1\]
\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(3 + a\right)\right)\right) - 1
↓
\left(\left(\left(b \cdot b\right) \cdot 12 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3} \cdot \sqrt{a \cdot a + b \cdot b}\right) + 4 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right)\right) + -1
double f(double a, double b) {
        double r7735473 = a;
        double r7735474 = r7735473 * r7735473;
        double r7735475 = b;
        double r7735476 = r7735475 * r7735475;
        double r7735477 = r7735474 + r7735476;
        double r7735478 = 2.0;
        double r7735479 = pow(r7735477, r7735478);
        double r7735480 = 4.0;
        double r7735481 = 1.0;
        double r7735482 = r7735481 - r7735473;
        double r7735483 = r7735474 * r7735482;
        double r7735484 = 3.0;
        double r7735485 = r7735484 + r7735473;
        double r7735486 = r7735476 * r7735485;
        double r7735487 = r7735483 + r7735486;
        double r7735488 = r7735480 * r7735487;
        double r7735489 = r7735479 + r7735488;
        double r7735490 = r7735489 - r7735481;
        return r7735490;
}
↓
double f(double a, double b) {
        double r7735491 = b;
        double r7735492 = r7735491 * r7735491;
        double r7735493 = 12.0;
        double r7735494 = r7735492 * r7735493;
        double r7735495 = a;
        double r7735496 = r7735495 * r7735495;
        double r7735497 = r7735496 + r7735492;
        double r7735498 = sqrt(r7735497);
        double r7735499 = 3.0;
        double r7735500 = pow(r7735498, r7735499);
        double r7735501 = r7735500 * r7735498;
        double r7735502 = r7735494 + r7735501;
        double r7735503 = 4.0;
        double r7735504 = r7735492 - r7735496;
        double r7735505 = r7735504 * r7735495;
        double r7735506 = r7735505 + r7735496;
        double r7735507 = r7735503 * r7735506;
        double r7735508 = r7735502 + r7735507;
        double r7735509 = -1.0;
        double r7735510 = r7735508 + r7735509;
        return r7735510;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\left(\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) + \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right) \cdot 4\right) + -1}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \left(\left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) + \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right) \cdot 4\right) + -1\]

  5. Applied associate-*l*0.1

    \[\leadsto \left(\left(\color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(a \cdot a + b \cdot b\right)\right)} + \left(b \cdot b\right) \cdot 12\right) + \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right) \cdot 4\right) + -1\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

    \[\leadsto \left(\left(\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}\right) + \left(b \cdot b\right) \cdot 12\right) + \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right) \cdot 4\right) + -1\]

  8. Applied cube-unmult0.1

    \[\leadsto \left(\left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3}} + \left(b \cdot b\right) \cdot 12\right) + \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right) \cdot 4\right) + -1\]

  9. Final simplification0.1

    \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 12 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{3} \cdot \sqrt{a \cdot a + b \cdot b}\right) + 4 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot a + a \cdot a\right)\right) + -1\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))