The quadratic formula (r1)

Average Error: 33.2 → 6.7

Time: 20.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3798449810939068 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{\frac{b}{a}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 2.051368100893223 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 1.2859619246531207 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} + b} \cdot \left(-4 \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{2 \cdot b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3897295 = b;
        double r3897296 = -r3897295;
        double r3897297 = r3897295 * r3897295;
        double r3897298 = 4.0;
        double r3897299 = a;
        double r3897300 = r3897298 * r3897299;
        double r3897301 = c;
        double r3897302 = r3897300 * r3897301;
        double r3897303 = r3897297 - r3897302;
        double r3897304 = sqrt(r3897303);
        double r3897305 = r3897296 + r3897304;
        double r3897306 = 2.0;
        double r3897307 = r3897306 * r3897299;
        double r3897308 = r3897305 / r3897307;
        return r3897308;
}

↓

double f(double a, double b, double c) {
        double r3897309 = b;
        double r3897310 = -1.3798449810939068e+76;
        bool r3897311 = r3897309 <= r3897310;
        double r3897312 = c;
        double r3897313 = a;
        double r3897314 = r3897309 / r3897313;
        double r3897315 = r3897312 / r3897314;
        double r3897316 = r3897315 - r3897309;
        double r3897317 = 2.0;
        double r3897318 = r3897316 * r3897317;
        double r3897319 = r3897318 / r3897313;
        double r3897320 = r3897319 / r3897317;
        double r3897321 = 2.051368100893223e-286;
        bool r3897322 = r3897309 <= r3897321;
        double r3897323 = -4.0;
        double r3897324 = r3897323 * r3897312;
        double r3897325 = r3897309 * r3897309;
        double r3897326 = fma(r3897324, r3897313, r3897325);
        double r3897327 = sqrt(r3897326);
        double r3897328 = r3897327 - r3897309;
        double r3897329 = sqrt(r3897328);
        double r3897330 = r3897313 / r3897329;
        double r3897331 = r3897329 / r3897330;
        double r3897332 = r3897331 / r3897317;
        double r3897333 = 1.2859619246531207e+138;
        bool r3897334 = r3897309 <= r3897333;
        double r3897335 = 0.5;
        double r3897336 = fma(r3897313, r3897324, r3897325);
        double r3897337 = sqrt(r3897336);
        double r3897338 = r3897337 + r3897309;
        double r3897339 = r3897335 / r3897338;
        double r3897340 = r3897339 * r3897324;
        double r3897341 = r3897317 * r3897309;
        double r3897342 = r3897324 / r3897341;
        double r3897343 = r3897342 / r3897317;
        double r3897344 = r3897334 ? r3897340 : r3897343;
        double r3897345 = r3897322 ? r3897332 : r3897344;
        double r3897346 = r3897311 ? r3897320 : r3897345;
        return r3897346;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.2
Target	20.2
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -1.3798449810939068e+76

   1. Initial program 39.5

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

   2. Simplified 39.5

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

   3. Taylor expanded around -inf 9.8

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

   4. Simplified 4.4

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

   if -1.3798449810939068e+76 < b < 2.051368100893223e-286

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   3. Taylor expanded around 0 8.9

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

   4. Simplified 8.9

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

   6. Applied add-sqr-sqrt 9.2

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

   7. Applied associate-/l* 9.2

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

   if 2.051368100893223e-286 < b < 1.2859619246531207e+138

   1. Initial program 34.9

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

   2. Simplified 34.8

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

   3. Taylor expanded around 0 34.8

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

   4. Simplified 34.9

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

   6. Applied flip-- 35.0

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

   7. Simplified 16.5

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

   9. Applied *-un-lft-identity 16.5

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

   10. Applied *-un-lft-identity 16.5

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

   11. Applied distribute-lft-out 16.5

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

   12. Applied times-frac 15.7

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

   13. Applied associate-/l* 10.9

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

   14. Simplified 8.3

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

   16. Applied *-un-lft-identity 8.3

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

   17. Applied *-un-lft-identity 8.3

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

   18. Applied *-un-lft-identity 8.3

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

   19. Applied distribute-lft-out 8.3

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

   20. Applied div-inv 8.3

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

   21. Applied times-frac 8.4

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

   22. Applied times-frac 8.4

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

   23. Simplified 8.4

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

   24. Simplified 8.4

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

   if 1.2859619246531207e+138 < b

   1. Initial program 61.2

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

   2. Simplified 61.2

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

   3. Taylor expanded around 0 61.2

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

   4. Simplified 61.2

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

   6. Applied flip-- 61.2

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

   7. Simplified 35.1

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

   9. Applied *-un-lft-identity 35.1

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

   10. Applied *-un-lft-identity 35.1

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

   11. Applied distribute-lft-out 35.1

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

   12. Applied times-frac 35.4

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

   13. Applied associate-/l* 35.2

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

   14. Simplified 34.0

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

   15. Taylor expanded around 0 1.8

       \[\leadsto \frac{\frac{\frac{c \cdot -4}{1}}{\color{blue}{2 \cdot b}}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3798449810939068 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{\frac{b}{a}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 2.051368100893223 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 1.2859619246531207 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} + b} \cdot \left(-4 \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{2 \cdot b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))