quadm (p42, negative)

Average Error: 34.3 → 9.8

Time: 15.7s

Precision: 64

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.721169858947977983937490110883363055351 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{0 \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2}}{a} + \frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(\left(-4\right) \cdot c, a, {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1920982614230223.5:\\ \;\;\;\;\frac{\frac{0 \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2}}{a} + \frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r53632 = b;
        double r53633 = -r53632;
        double r53634 = r53632 * r53632;
        double r53635 = 4.0;
        double r53636 = a;
        double r53637 = c;
        double r53638 = r53636 * r53637;
        double r53639 = r53635 * r53638;
        double r53640 = r53634 - r53639;
        double r53641 = sqrt(r53640);
        double r53642 = r53633 - r53641;
        double r53643 = 2.0;
        double r53644 = r53643 * r53636;
        double r53645 = r53642 / r53644;
        return r53645;
}

↓

double f(double a, double b, double c) {
        double r53646 = b;
        double r53647 = -7.359940312872037e+54;
        bool r53648 = r53646 <= r53647;
        double r53649 = -1.0;
        double r53650 = c;
        double r53651 = r53650 / r53646;
        double r53652 = r53649 * r53651;
        double r53653 = -3.721169858947978e-153;
        bool r53654 = r53646 <= r53653;
        double r53655 = 0.0;
        double r53656 = r53646 * r53646;
        double r53657 = a;
        double r53658 = r53657 * r53650;
        double r53659 = 4.0;
        double r53660 = r53658 * r53659;
        double r53661 = r53656 - r53660;
        double r53662 = sqrt(r53661);
        double r53663 = r53655 * r53662;
        double r53664 = 2.0;
        double r53665 = r53663 / r53664;
        double r53666 = r53665 / r53657;
        double r53667 = r53659 * r53658;
        double r53668 = -r53659;
        double r53669 = r53668 * r53650;
        double r53670 = 2.0;
        double r53671 = pow(r53646, r53670);
        double r53672 = fma(r53669, r53657, r53671);
        double r53673 = sqrt(r53672);
        double r53674 = r53673 - r53646;
        double r53675 = r53667 / r53674;
        double r53676 = r53664 * r53657;
        double r53677 = r53675 / r53676;
        double r53678 = r53666 + r53677;
        double r53679 = 1920982614230223.5;
        bool r53680 = r53646 <= r53679;
        double r53681 = -r53646;
        double r53682 = r53681 - r53662;
        double r53683 = r53682 / r53676;
        double r53684 = r53666 + r53683;
        double r53685 = 1.0;
        double r53686 = r53646 / r53657;
        double r53687 = r53651 - r53686;
        double r53688 = r53685 * r53687;
        double r53689 = r53680 ? r53684 : r53688;
        double r53690 = r53654 ? r53678 : r53689;
        double r53691 = r53648 ? r53652 : r53690;
        return r53691;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.3
Target	20.9
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -7.359940312872037e+54

   1. Initial program 57.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Taylor expanded around -inf 3.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   if -7.359940312872037e+54 < b < -3.721169858947978e-153

   1. Initial program 39.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied clear-num 39.7

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

   5. Applied div-inv 39.7

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

   6. Applied add-cube-cbrt 39.7

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

   7. Applied times-frac 39.7

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

   8. Simplified 39.7

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

   9. Simplified 39.7

      \[\leadsto \frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 41.1

       \[\leadsto \frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}}\right)\]

   12. Applied add-sqr-sqrt 41.1

       \[\leadsto \frac{1}{2 \cdot a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\]

   13. Applied prod-diff 41.4

       \[\leadsto \frac{1}{2 \cdot a} \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, -\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)\right)}\]

   14. Applied distribute-lft-in 41.4

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, -\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right) + \frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)}\]

   15. Simplified 40.9

       \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}} + \frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)\]

   16. Simplified 39.6

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

   18. Applied flip-- 39.7

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

   19. Simplified 18.5

       \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a} + \frac{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} \cdot \left(-1 + 1\right)}{2}}{a}\]

   20. Simplified 18.6

       \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(\left(-4\right) \cdot c, a, {b}^{2}\right)} - b}}}{2 \cdot a} + \frac{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} \cdot \left(-1 + 1\right)}{2}}{a}\]

   if -3.721169858947978e-153 < b < 1920982614230223.5

   1. Initial program 12.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied clear-num 12.4

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

   5. Applied div-inv 12.5

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

   6. Applied add-cube-cbrt 12.5

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

   7. Applied times-frac 12.5

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

   8. Simplified 12.5

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

   9. Simplified 12.4

      \[\leadsto \frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 13.1

       \[\leadsto \frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}}\right)\]

   12. Applied add-sqr-sqrt 50.2

       \[\leadsto \frac{1}{2 \cdot a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\]

   13. Applied prod-diff 50.2

       \[\leadsto \frac{1}{2 \cdot a} \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, -\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)\right)}\]

   14. Applied distribute-lft-in 50.2

       \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, -\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right) + \frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)}\]

   15. Simplified 12.3

       \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}} + \frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}, \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}\right)\right)\]

   16. Simplified 12.3

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

   if 1920982614230223.5 < b

   1. Initial program 34.0

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

   3. Simplified 6.8

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.721169858947977983937490110883363055351 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{0 \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2}}{a} + \frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(\left(-4\right) \cdot c, a, {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1920982614230223.5:\\ \;\;\;\;\frac{\frac{0 \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2}}{a} + \frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))