quadp (p42, positive)

Average Error: 33.8 → 9.7

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5568198144016078 \cdot 10^{-163}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 40732783.803636283:\\ \;\;\;\;1 \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r73313 = b;
        double r73314 = -r73313;
        double r73315 = r73313 * r73313;
        double r73316 = 4.0;
        double r73317 = a;
        double r73318 = c;
        double r73319 = r73317 * r73318;
        double r73320 = r73316 * r73319;
        double r73321 = r73315 - r73320;
        double r73322 = sqrt(r73321);
        double r73323 = r73314 + r73322;
        double r73324 = 2.0;
        double r73325 = r73324 * r73317;
        double r73326 = r73323 / r73325;
        return r73326;
}

↓

double f(double a, double b, double c) {
        double r73327 = b;
        double r73328 = -7.700313305414632e+138;
        bool r73329 = r73327 <= r73328;
        double r73330 = 1.0;
        double r73331 = c;
        double r73332 = r73331 / r73327;
        double r73333 = a;
        double r73334 = r73327 / r73333;
        double r73335 = r73332 - r73334;
        double r73336 = r73330 * r73335;
        double r73337 = 1.5568198144016078e-163;
        bool r73338 = r73327 <= r73337;
        double r73339 = 1.0;
        double r73340 = -r73327;
        double r73341 = r73327 * r73327;
        double r73342 = 4.0;
        double r73343 = r73333 * r73331;
        double r73344 = r73342 * r73343;
        double r73345 = r73341 - r73344;
        double r73346 = sqrt(r73345);
        double r73347 = r73340 + r73346;
        double r73348 = 2.0;
        double r73349 = r73348 * r73333;
        double r73350 = r73347 / r73349;
        double r73351 = r73339 * r73350;
        double r73352 = 40732783.80363628;
        bool r73353 = r73327 <= r73352;
        double r73354 = 0.0;
        double r73355 = r73354 + r73344;
        double r73356 = r73340 - r73346;
        double r73357 = r73349 * r73356;
        double r73358 = r73355 / r73357;
        double r73359 = r73339 * r73358;
        double r73360 = -1.0;
        double r73361 = r73360 * r73332;
        double r73362 = r73353 ? r73359 : r73361;
        double r73363 = r73338 ? r73351 : r73362;
        double r73364 = r73329 ? r73336 : r73363;
        return r73364;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	21.1
Herbie	9.7

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -7.700313305414632e+138

   1. Initial program 57.3

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

   2. Taylor expanded around -inf 2.9

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

   3. Simplified 2.9

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

   if -7.700313305414632e+138 < b < 1.5568198144016078e-163

   1. Initial program 10.6

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

   3. Applied clear-num 10.7

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

   5. Applied *-un-lft-identity 10.7

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

   6. Applied add-cube-cbrt 10.7

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

   7. Applied times-frac 10.7

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

   8. Simplified 10.7

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

   9. Simplified 10.6

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

   if 1.5568198144016078e-163 < b < 40732783.80363628

   1. Initial program 32.3

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

   3. Applied clear-num 32.3

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

   5. Applied *-un-lft-identity 32.3

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

   6. Applied add-cube-cbrt 32.3

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

   7. Applied times-frac 32.3

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

   8. Simplified 32.3

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

   9. Simplified 32.3

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

   11. Applied flip-+ 32.3

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

   12. Simplified 17.3

       \[\leadsto 1 \cdot \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   13. Using strategy rm

   14. Applied div-inv 17.3

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

   15. Applied associate-/l* 22.6

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

   16. Simplified 22.5

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

   if 40732783.80363628 < b

   1. Initial program 55.9

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

   2. Taylor expanded around inf 6.0

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5568198144016078 \cdot 10^{-163}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 40732783.803636283:\\ \;\;\;\;1 \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))