Average Error: 0.2 → 0.2
Time: 39.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(-1 + \sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)} \cdot \left(\left(\sqrt{\sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)}} \cdot a\right) \cdot \left(\sqrt{\sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)}} \cdot a\right)\right)\right) + b \cdot \left(\left(\left(12 + 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)\right) \cdot b\right)\]
\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(-1 + \sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)} \cdot \left(\left(\sqrt{\sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)}} \cdot a\right) \cdot \left(\sqrt{\sqrt{\left(4 - 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)}} \cdot a\right)\right)\right) + b \cdot \left(\left(\left(12 + 4 \cdot a\right) + \left(b \cdot b + a \cdot a\right)\right) \cdot b\right)
double f(double a, double b) {
        double r9173344 = a;
        double r9173345 = r9173344 * r9173344;
        double r9173346 = b;
        double r9173347 = r9173346 * r9173346;
        double r9173348 = r9173345 + r9173347;
        double r9173349 = 2.0;
        double r9173350 = pow(r9173348, r9173349);
        double r9173351 = 4.0;
        double r9173352 = 1.0;
        double r9173353 = r9173352 - r9173344;
        double r9173354 = r9173345 * r9173353;
        double r9173355 = 3.0;
        double r9173356 = r9173355 + r9173344;
        double r9173357 = r9173347 * r9173356;
        double r9173358 = r9173354 + r9173357;
        double r9173359 = r9173351 * r9173358;
        double r9173360 = r9173350 + r9173359;
        double r9173361 = r9173360 - r9173352;
        return r9173361;
}
↓
double f(double a, double b) {
        double r9173362 = -1.0;
        double r9173363 = 4.0;
        double r9173364 = a;
        double r9173365 = r9173363 * r9173364;
        double r9173366 = r9173363 - r9173365;
        double r9173367 = b;
        double r9173368 = r9173367 * r9173367;
        double r9173369 = r9173364 * r9173364;
        double r9173370 = r9173368 + r9173369;
        double r9173371 = r9173366 + r9173370;
        double r9173372 = sqrt(r9173371);
        double r9173373 = sqrt(r9173372);
        double r9173374 = r9173373 * r9173364;
        double r9173375 = r9173374 * r9173374;
        double r9173376 = r9173372 * r9173375;
        double r9173377 = r9173362 + r9173376;
        double r9173378 = 12.0;
        double r9173379 = r9173378 + r9173365;
        double r9173380 = r9173379 + r9173370;
        double r9173381 = r9173380 * r9173367;
        double r9173382 = r9173367 * r9173381;
        double r9173383 = r9173377 + r9173382;
        return r9173383;
}

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

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

  4. Applied associate-*l*0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied associate-*r*0.1

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

  9. Applied add-sqr-sqrt0.1

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

  10. Applied sqrt-prod0.2

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

  11. Applied unswap-sqr0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 
(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))