Average Error: 0.2 → 0.0
Time: 1.1m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right) + 4 \cdot \left(a \cdot \left(a \cdot a + a\right) + \left(b \cdot \left(1 - 3 \cdot a\right)\right) \cdot b\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} - 1\right) + 4 \cdot \left(a \cdot \left(a \cdot a + a\right) + \left(b \cdot \left(1 - 3 \cdot a\right)\right) \cdot b\right)
double f(double a, double b) {
        double r72764566 = a;
        double r72764567 = r72764566 * r72764566;
        double r72764568 = b;
        double r72764569 = r72764568 * r72764568;
        double r72764570 = r72764567 + r72764569;
        double r72764571 = 2.0;
        double r72764572 = pow(r72764570, r72764571);
        double r72764573 = 4.0;
        double r72764574 = 1.0;
        double r72764575 = r72764574 + r72764566;
        double r72764576 = r72764567 * r72764575;
        double r72764577 = 3.0;
        double r72764578 = r72764577 * r72764566;
        double r72764579 = r72764574 - r72764578;
        double r72764580 = r72764569 * r72764579;
        double r72764581 = r72764576 + r72764580;
        double r72764582 = r72764573 * r72764581;
        double r72764583 = r72764572 + r72764582;
        double r72764584 = r72764583 - r72764574;
        return r72764584;
}
↓
double f(double a, double b) {
        double r72764585 = a;
        double r72764586 = r72764585 * r72764585;
        double r72764587 = b;
        double r72764588 = r72764587 * r72764587;
        double r72764589 = r72764586 + r72764588;
        double r72764590 = sqrt(r72764589);
        double r72764591 = 4.0;
        double r72764592 = pow(r72764590, r72764591);
        double r72764593 = 1.0;
        double r72764594 = r72764592 - r72764593;
        double r72764595 = r72764586 + r72764585;
        double r72764596 = r72764585 * r72764595;
        double r72764597 = 3.0;
        double r72764598 = r72764597 * r72764585;
        double r72764599 = r72764593 - r72764598;
        double r72764600 = r72764587 * r72764599;
        double r72764601 = r72764600 * r72764587;
        double r72764602 = r72764596 + r72764601;
        double r72764603 = r72764591 * r72764602;
        double r72764604 = r72764594 + r72764603;
        return r72764604;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 
(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))