Average Error: 0.2 → 0.1
Time: 58.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4 + \left(\left(b \cdot \left(2 \cdot \left(a \cdot a\right) + b \cdot b\right)\right) \cdot b + {a}^{4}\right)\right) - 1\]
\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(\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot 4 + \left(\left(b \cdot \left(2 \cdot \left(a \cdot a\right) + b \cdot b\right)\right) \cdot b + {a}^{4}\right)\right) - 1
double f(double a, double b) {
        double r48144372 = a;
        double r48144373 = r48144372 * r48144372;
        double r48144374 = b;
        double r48144375 = r48144374 * r48144374;
        double r48144376 = r48144373 + r48144375;
        double r48144377 = 2.0;
        double r48144378 = pow(r48144376, r48144377);
        double r48144379 = 4.0;
        double r48144380 = 1.0;
        double r48144381 = r48144380 + r48144372;
        double r48144382 = r48144373 * r48144381;
        double r48144383 = 3.0;
        double r48144384 = r48144383 * r48144372;
        double r48144385 = r48144380 - r48144384;
        double r48144386 = r48144375 * r48144385;
        double r48144387 = r48144382 + r48144386;
        double r48144388 = r48144379 * r48144387;
        double r48144389 = r48144378 + r48144388;
        double r48144390 = r48144389 - r48144380;
        return r48144390;
}
↓
double f(double a, double b) {
        double r48144391 = 1.0;
        double r48144392 = 3.0;
        double r48144393 = a;
        double r48144394 = r48144392 * r48144393;
        double r48144395 = r48144391 - r48144394;
        double r48144396 = b;
        double r48144397 = r48144396 * r48144396;
        double r48144398 = r48144395 * r48144397;
        double r48144399 = r48144393 * r48144393;
        double r48144400 = r48144391 + r48144393;
        double r48144401 = r48144399 * r48144400;
        double r48144402 = r48144398 + r48144401;
        double r48144403 = 4.0;
        double r48144404 = r48144402 * r48144403;
        double r48144405 = 2.0;
        double r48144406 = r48144405 * r48144399;
        double r48144407 = r48144406 + r48144397;
        double r48144408 = r48144396 * r48144407;
        double r48144409 = r48144408 * r48144396;
        double r48144410 = pow(r48144393, r48144403);
        double r48144411 = r48144409 + r48144410;
        double r48144412 = r48144404 + r48144411;
        double r48144413 = r48144412 - r48144391;
        return r48144413;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.2

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

  5. Applied associate-*r*0.2

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

  7. Applied pow10.2

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

  8. Applied pow10.2

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

  9. Applied pow-prod-up0.2

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

  10. Applied pow10.2

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

  11. Applied pow10.2

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

  12. Applied pow-prod-up0.2

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

  13. Applied pow-prod-up0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 
(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))