quadm (p42, negative)

Average Error: 33.1 → 6.7

Time: 3.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.9239515900644342 \cdot 10^{+146}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.1748628172197204 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}\\ \mathbf{elif}\;b \le 2.0710701119913226 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r15890345 = b;
        double r15890346 = -r15890345;
        double r15890347 = r15890345 * r15890345;
        double r15890348 = 4.0;
        double r15890349 = a;
        double r15890350 = c;
        double r15890351 = r15890349 * r15890350;
        double r15890352 = r15890348 * r15890351;
        double r15890353 = r15890347 - r15890352;
        double r15890354 = sqrt(r15890353);
        double r15890355 = r15890346 - r15890354;
        double r15890356 = 2.0;
        double r15890357 = r15890356 * r15890349;
        double r15890358 = r15890355 / r15890357;
        return r15890358;
}

↓

double f(double a, double b, double c) {
        double r15890359 = b;
        double r15890360 = -1.9239515900644342e+146;
        bool r15890361 = r15890359 <= r15890360;
        double r15890362 = c;
        double r15890363 = r15890362 / r15890359;
        double r15890364 = -r15890363;
        double r15890365 = 1.1748628172197204e-280;
        bool r15890366 = r15890359 <= r15890365;
        double r15890367 = 0.5;
        double r15890368 = r15890362 / r15890367;
        double r15890369 = a;
        double r15890370 = r15890362 * r15890369;
        double r15890371 = -4.0;
        double r15890372 = r15890359 * r15890359;
        double r15890373 = fma(r15890370, r15890371, r15890372);
        double r15890374 = sqrt(r15890373);
        double r15890375 = r15890374 - r15890359;
        double r15890376 = r15890368 / r15890375;
        double r15890377 = 2.0710701119913226e+79;
        bool r15890378 = r15890359 <= r15890377;
        double r15890379 = -r15890359;
        double r15890380 = r15890379 - r15890374;
        double r15890381 = 2.0;
        double r15890382 = r15890381 * r15890369;
        double r15890383 = r15890380 / r15890382;
        double r15890384 = r15890359 / r15890369;
        double r15890385 = r15890363 - r15890384;
        double r15890386 = r15890378 ? r15890383 : r15890385;
        double r15890387 = r15890366 ? r15890376 : r15890386;
        double r15890388 = r15890361 ? r15890364 : r15890387;
        return r15890388;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.1
Target	20.4
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -1.9239515900644342e+146

   1. Initial program 62.0

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

   2. Simplified 62.0

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

   4. Applied associate-/l/ 62.0

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

   5. Taylor expanded around -inf 1.7

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

   6. Simplified 1.7

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

   if -1.9239515900644342e+146 < b < 1.1748628172197204e-280

   1. Initial program 33.3

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

   2. Simplified 33.3

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

   4. Applied flip-- 33.4

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

   5. Simplified 16.0

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

   6. Simplified 16.0

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

   8. Applied *-un-lft-identity 16.0

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

   9. Applied *-un-lft-identity 16.0

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

   10. Applied *-un-lft-identity 16.0

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

   11. Applied times-frac 16.0

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

   12. Applied times-frac 16.0

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

   13. Simplified 16.0

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

   14. Simplified 8.7

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

   if 1.1748628172197204e-280 < b < 2.0710701119913226e+79

   1. Initial program 8.5

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

   2. Simplified 8.5

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

   4. Applied associate-/l/ 8.5

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

   if 2.0710701119913226e+79 < b

   1. Initial program 40.5

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

   2. Simplified 40.4

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

   3. Taylor expanded around inf 4.7

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9239515900644342 \cdot 10^{+146}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.1748628172197204 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}\\ \mathbf{elif}\;b \le 2.0710701119913226 \cdot 10^{+79}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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)))