quadm (p42, negative)

Average Error: 33.2 → 9.9

Time: 30.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1191214 = b;
        double r1191215 = -r1191214;
        double r1191216 = r1191214 * r1191214;
        double r1191217 = 4.0;
        double r1191218 = a;
        double r1191219 = c;
        double r1191220 = r1191218 * r1191219;
        double r1191221 = r1191217 * r1191220;
        double r1191222 = r1191216 - r1191221;
        double r1191223 = sqrt(r1191222);
        double r1191224 = r1191215 - r1191223;
        double r1191225 = 2.0;
        double r1191226 = r1191225 * r1191218;
        double r1191227 = r1191224 / r1191226;
        return r1191227;
}

↓

double f(double a, double b, double c) {
        double r1191228 = b;
        double r1191229 = -1.8774910265390396e-73;
        bool r1191230 = r1191228 <= r1191229;
        double r1191231 = -2.0;
        double r1191232 = c;
        double r1191233 = r1191232 / r1191228;
        double r1191234 = r1191231 * r1191233;
        double r1191235 = 2.0;
        double r1191236 = r1191234 / r1191235;
        double r1191237 = 2.5703497435733685e+102;
        bool r1191238 = r1191228 <= r1191237;
        double r1191239 = 1.0;
        double r1191240 = -r1191228;
        double r1191241 = a;
        double r1191242 = -4.0;
        double r1191243 = r1191241 * r1191242;
        double r1191244 = r1191228 * r1191228;
        double r1191245 = fma(r1191243, r1191232, r1191244);
        double r1191246 = sqrt(r1191245);
        double r1191247 = r1191240 - r1191246;
        double r1191248 = r1191247 / r1191241;
        double r1191249 = r1191239 / r1191248;
        double r1191250 = r1191239 / r1191249;
        double r1191251 = r1191250 / r1191235;
        double r1191252 = r1191228 / r1191241;
        double r1191253 = r1191233 - r1191252;
        double r1191254 = r1191253 * r1191235;
        double r1191255 = r1191254 / r1191235;
        double r1191256 = r1191238 ? r1191251 : r1191255;
        double r1191257 = r1191230 ? r1191236 : r1191256;
        return r1191257;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.2
Target	20.4
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if b < -1.8774910265390396e-73

   1. Initial program 52.5

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

   2. Simplified 52.5

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

   4. Applied div-inv 52.5

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

   5. Taylor expanded around -inf 8.6

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

   if -1.8774910265390396e-73 < b < 2.5703497435733685e+102

   1. Initial program 13.1

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

   2. Simplified 13.1

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

   4. Applied *-un-lft-identity 13.1

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

   5. Applied *-un-lft-identity 13.1

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

   6. Applied distribute-rgt-neg-in 13.1

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

   7. Applied distribute-lft-out-- 13.1

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

   8. Applied associate-/l* 13.2

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

   10. Applied *-un-lft-identity 13.2

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

   11. Applied associate-/l* 13.2

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

   if 2.5703497435733685e+102 < b

   1. Initial program 43.9

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

   2. Simplified 43.9

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

   4. Applied *-un-lft-identity 43.9

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

   5. Applied *-un-lft-identity 43.9

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

   6. Applied distribute-rgt-neg-in 43.9

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

   7. Applied distribute-lft-out-- 43.9

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

   8. Applied associate-/l* 43.9

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

   10. Applied *-un-lft-identity 43.9

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

   11. Applied associate-/l* 43.9

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

   12. Taylor expanded around inf 3.0

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

   13. Simplified 3.0

       \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

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