quadp (p42, positive)

Average Error: 33.8 → 10.1

Time: 8.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r75685 = b;
        double r75686 = -r75685;
        double r75687 = r75685 * r75685;
        double r75688 = 4.0;
        double r75689 = a;
        double r75690 = c;
        double r75691 = r75689 * r75690;
        double r75692 = r75688 * r75691;
        double r75693 = r75687 - r75692;
        double r75694 = sqrt(r75693);
        double r75695 = r75686 + r75694;
        double r75696 = 2.0;
        double r75697 = r75696 * r75689;
        double r75698 = r75695 / r75697;
        return r75698;
}

↓

double f(double a, double b, double c) {
        double r75699 = b;
        double r75700 = -3.124283374205192e+57;
        bool r75701 = r75699 <= r75700;
        double r75702 = -2.0;
        double r75703 = r75702 * r75699;
        double r75704 = 2.0;
        double r75705 = r75703 / r75704;
        double r75706 = a;
        double r75707 = r75705 / r75706;
        double r75708 = 3.84613441880261e-81;
        bool r75709 = r75699 <= r75708;
        double r75710 = 4.0;
        double r75711 = c;
        double r75712 = r75706 * r75711;
        double r75713 = r75710 * r75712;
        double r75714 = -r75713;
        double r75715 = fma(r75699, r75699, r75714);
        double r75716 = sqrt(r75715);
        double r75717 = r75716 - r75699;
        double r75718 = r75717 / r75704;
        double r75719 = r75718 / r75706;
        double r75720 = -1.0;
        double r75721 = r75711 / r75699;
        double r75722 = r75720 * r75721;
        double r75723 = r75709 ? r75719 : r75722;
        double r75724 = r75701 ? r75707 : r75723;
        return r75724;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.4
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if b < -3.124283374205192e+57

   1. Initial program 39.5

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

   2. Simplified 39.5

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

   4. Applied *-un-lft-identity 39.5

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

   5. Applied div-inv 39.5

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

   6. Applied times-frac 39.6

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

   7. Simplified 39.6

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

   9. Applied *-un-lft-identity 39.6

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

   10. Applied associate-*l* 39.6

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

   11. Simplified 39.5

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

   13. Applied add-cube-cbrt 39.6

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

   14. Applied sqrt-prod 39.6

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

   15. Applied fma-neg 39.6

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

   16. Taylor expanded around -inf 5.6

       \[\leadsto 1 \cdot \frac{\frac{\color{blue}{-2 \cdot b}}{2}}{a}\]

   17. Simplified 5.6

       \[\leadsto 1 \cdot \frac{\frac{\color{blue}{-2 \cdot b}}{2}}{a}\]

   if -3.124283374205192e+57 < b < 3.84613441880261e-81

   1. Initial program 12.7

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

   2. Simplified 12.7

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

   4. Applied *-un-lft-identity 12.7

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

   5. Applied div-inv 12.7

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

   6. Applied times-frac 12.8

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

   7. Simplified 12.8

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

   9. Applied *-un-lft-identity 12.8

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

   10. Applied associate-*l* 12.8

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

   11. Simplified 12.7

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

   if 3.84613441880261e-81 < b

   1. Initial program 53.0

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

   2. Simplified 53.0

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

   4. Applied *-un-lft-identity 53.0

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

   5. Applied div-inv 53.0

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

   6. Applied times-frac 53.0

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

   7. Simplified 53.0

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

   9. Applied *-un-lft-identity 53.0

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

   10. Applied associate-*l* 53.0

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

   11. Simplified 53.0

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

   12. Taylor expanded around inf 9.5

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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