quadp (p42, positive)

Average Error: 32.9 → 6.6

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.295892693885355 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 8.470092598943746 \cdot 10^{+147}:\\ \;\;\;\;\left(c \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(-4 \cdot a\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot c\right), a, \left(b \cdot b\right)\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r9033206 = b;
        double r9033207 = -r9033206;
        double r9033208 = r9033206 * r9033206;
        double r9033209 = 4.0;
        double r9033210 = a;
        double r9033211 = c;
        double r9033212 = r9033210 * r9033211;
        double r9033213 = r9033209 * r9033212;
        double r9033214 = r9033208 - r9033213;
        double r9033215 = sqrt(r9033214);
        double r9033216 = r9033207 + r9033215;
        double r9033217 = 2.0;
        double r9033218 = r9033217 * r9033210;
        double r9033219 = r9033216 / r9033218;
        return r9033219;
}

↓

double f(double a, double b, double c) {
        double r9033220 = b;
        double r9033221 = -4.295892693885355e+120;
        bool r9033222 = r9033220 <= r9033221;
        double r9033223 = c;
        double r9033224 = r9033223 / r9033220;
        double r9033225 = a;
        double r9033226 = r9033224 * r9033225;
        double r9033227 = r9033226 - r9033220;
        double r9033228 = 2.0;
        double r9033229 = r9033227 * r9033228;
        double r9033230 = r9033229 / r9033228;
        double r9033231 = r9033230 / r9033225;
        double r9033232 = 5.818192251940127e-227;
        bool r9033233 = r9033220 <= r9033232;
        double r9033234 = -4.0;
        double r9033235 = r9033234 * r9033223;
        double r9033236 = r9033220 * r9033220;
        double r9033237 = fma(r9033225, r9033235, r9033236);
        double r9033238 = sqrt(r9033237);
        double r9033239 = r9033238 - r9033220;
        double r9033240 = r9033228 * r9033225;
        double r9033241 = r9033239 / r9033240;
        double r9033242 = 8.470092598943746e+147;
        bool r9033243 = r9033220 <= r9033242;
        double r9033244 = 0.5;
        double r9033245 = r9033244 / r9033225;
        double r9033246 = r9033234 * r9033225;
        double r9033247 = r9033245 * r9033246;
        double r9033248 = r9033223 * r9033247;
        double r9033249 = 1.0;
        double r9033250 = fma(r9033235, r9033225, r9033236);
        double r9033251 = sqrt(r9033250);
        double r9033252 = r9033251 + r9033220;
        double r9033253 = r9033249 / r9033252;
        double r9033254 = r9033248 * r9033253;
        double r9033255 = -r9033220;
        double r9033256 = r9033255 / r9033223;
        double r9033257 = r9033249 / r9033256;
        double r9033258 = r9033243 ? r9033254 : r9033257;
        double r9033259 = r9033233 ? r9033241 : r9033258;
        double r9033260 = r9033222 ? r9033231 : r9033259;
        return r9033260;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.9
Target	20.3
Herbie	6.6

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -4.295892693885355e+120

   1. Initial program 49.1

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

   2. Simplified 49.1

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

   3. Taylor expanded around -inf 10.2

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

   4. Simplified 2.9

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

   if -4.295892693885355e+120 < b < 5.818192251940127e-227

   1. Initial program 9.3

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

   2. Simplified 9.3

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

   4. Applied add-cube-cbrt 9.6

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

   5. Applied *-un-lft-identity 9.6

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

   6. Applied sqrt-prod 9.6

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

   7. Applied prod-diff 9.6

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

   8. Simplified 9.3

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

   9. Simplified 9.3

      \[\leadsto \frac{\frac{\left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied associate-/l/ 9.3

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

   12. Simplified 9.3

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

   if 5.818192251940127e-227 < b < 8.470092598943746e+147

   1. Initial program 36.3

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

   2. Simplified 36.3

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

   4. Applied add-cube-cbrt 39.9

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

   5. Applied *-un-lft-identity 39.9

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

   6. Applied sqrt-prod 39.9

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

   7. Applied prod-diff 40.7

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

   8. Simplified 40.7

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

   9. Simplified 36.3

      \[\leadsto \frac{\frac{\left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied associate-/l/ 36.3

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

   12. Simplified 36.3

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

   14. Applied clear-num 36.3

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

   16. Applied flip-- 36.4

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

   17. Applied associate-/r/ 36.5

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

   18. Applied add-sqr-sqrt 36.5

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

   19. Applied times-frac 36.5

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

   20. Simplified 7.3

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

   21. Simplified 7.3

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

   if 8.470092598943746e+147 < b

   1. Initial program 62.1

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

   2. Simplified 62.1

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

   4. Applied add-cube-cbrt 62.8

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

   5. Applied *-un-lft-identity 62.8

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

   6. Applied sqrt-prod 62.8

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

   7. Applied prod-diff 62.9

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

   8. Simplified 62.9

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

   9. Simplified 62.2

      \[\leadsto \frac{\frac{\left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right) + \color{blue}{0}}{2}}{a}\]
   10. Using strategy rm

   11. Applied associate-/l/ 62.2

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

   12. Simplified 62.2

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

   14. Applied clear-num 62.2

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

   15. Taylor expanded around 0 2.7

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

   16. Simplified 2.7

       \[\leadsto \frac{1}{\color{blue}{-\frac{b}{c}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.295892693885355 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 8.470092598943746 \cdot 10^{+147}:\\ \;\;\;\;\left(c \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(-4 \cdot a\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot c\right), a, \left(b \cdot b\right)\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))