quadp (p42, positive)

Average Error: 33.8 → 12.3

Time: 14.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.8410785388490295 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.2005525135011565 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{2 \cdot b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1601343 = b;
        double r1601344 = -r1601343;
        double r1601345 = r1601343 * r1601343;
        double r1601346 = 4.0;
        double r1601347 = a;
        double r1601348 = c;
        double r1601349 = r1601347 * r1601348;
        double r1601350 = r1601346 * r1601349;
        double r1601351 = r1601345 - r1601350;
        double r1601352 = sqrt(r1601351);
        double r1601353 = r1601344 + r1601352;
        double r1601354 = 2.0;
        double r1601355 = r1601354 * r1601347;
        double r1601356 = r1601353 / r1601355;
        return r1601356;
}

↓

double f(double a, double b, double c) {
        double r1601357 = b;
        double r1601358 = -1.8410785388490295e-298;
        bool r1601359 = r1601357 <= r1601358;
        double r1601360 = a;
        double r1601361 = c;
        double r1601362 = r1601360 * r1601361;
        double r1601363 = -4.0;
        double r1601364 = r1601357 * r1601357;
        double r1601365 = fma(r1601362, r1601363, r1601364);
        double r1601366 = sqrt(r1601365);
        double r1601367 = r1601366 / r1601360;
        double r1601368 = r1601357 / r1601360;
        double r1601369 = r1601367 - r1601368;
        double r1601370 = 2.0;
        double r1601371 = r1601369 / r1601370;
        double r1601372 = 3.2005525135011565e+83;
        bool r1601373 = r1601357 <= r1601372;
        double r1601374 = r1601363 * r1601361;
        double r1601375 = fma(r1601363, r1601362, r1601364);
        double r1601376 = sqrt(r1601375);
        double r1601377 = r1601376 + r1601357;
        double r1601378 = r1601374 / r1601377;
        double r1601379 = r1601378 / r1601370;
        double r1601380 = r1601370 * r1601357;
        double r1601381 = r1601374 / r1601380;
        double r1601382 = r1601381 / r1601370;
        double r1601383 = r1601373 ? r1601379 : r1601382;
        double r1601384 = r1601359 ? r1601371 : r1601383;
        return r1601384;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.3
Herbie	12.3

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

Derivation

1. Split input into 3 regimes

2. if b < -1.8410785388490295e-298

   1. Initial program 20.7

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

   2. Simplified 20.7

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

   4. Applied div-sub 20.7

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

   if -1.8410785388490295e-298 < b < 3.2005525135011565e+83

   1. Initial program 31.7

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

   2. Simplified 31.7

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

   4. Applied flip-- 31.9

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

   5. Simplified 16.4

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

   7. Applied *-un-lft-identity 16.4

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   8. Applied *-un-lft-identity 16.4

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}}{1 \cdot a}}{2}\]

   9. Applied *-un-lft-identity 16.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot c, -4, 0\right)}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}{1 \cdot a}}{2}\]

   10. Applied times-frac 16.4

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 16.4

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}}{2}\]

   12. Simplified 16.4

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]

   13. Simplified 15.8

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

   14. Taylor expanded around 0 8.3

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

   if 3.2005525135011565e+83 < b

   1. Initial program 57.9

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

   2. Simplified 57.9

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

   4. Applied flip-- 58.0

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

   5. Simplified 30.9

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

   7. Applied *-un-lft-identity 30.9

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   8. Applied *-un-lft-identity 30.9

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}}{1 \cdot a}}{2}\]

   9. Applied *-un-lft-identity 30.9

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot c, -4, 0\right)}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}{1 \cdot a}}{2}\]

   10. Applied times-frac 30.9

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 30.9

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}}{2}\]

   12. Simplified 30.9

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]

   13. Simplified 29.4

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

   14. Taylor expanded around 0 28.8

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

   15. Taylor expanded around 0 2.7

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

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8410785388490295 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.2005525135011565 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{2 \cdot b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))