quadp (p42, positive)

Average Error: 34.2 → 9.6

Time: 15.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3530991 = b;
        double r3530992 = -r3530991;
        double r3530993 = r3530991 * r3530991;
        double r3530994 = 4.0;
        double r3530995 = a;
        double r3530996 = c;
        double r3530997 = r3530995 * r3530996;
        double r3530998 = r3530994 * r3530997;
        double r3530999 = r3530993 - r3530998;
        double r3531000 = sqrt(r3530999);
        double r3531001 = r3530992 + r3531000;
        double r3531002 = 2.0;
        double r3531003 = r3531002 * r3530995;
        double r3531004 = r3531001 / r3531003;
        return r3531004;
}

↓

double f(double a, double b, double c) {
        double r3531005 = b;
        double r3531006 = -3.450829996567048e+138;
        bool r3531007 = r3531005 <= r3531006;
        double r3531008 = c;
        double r3531009 = r3531008 / r3531005;
        double r3531010 = a;
        double r3531011 = r3531005 / r3531010;
        double r3531012 = r3531009 - r3531011;
        double r3531013 = 1.0;
        double r3531014 = r3531012 * r3531013;
        double r3531015 = 4.626043257219638e-62;
        bool r3531016 = r3531005 <= r3531015;
        double r3531017 = 1.0;
        double r3531018 = 2.0;
        double r3531019 = r3531017 / r3531018;
        double r3531020 = r3531019 / r3531010;
        double r3531021 = r3531005 * r3531005;
        double r3531022 = 4.0;
        double r3531023 = r3531008 * r3531022;
        double r3531024 = r3531023 * r3531010;
        double r3531025 = r3531021 - r3531024;
        double r3531026 = sqrt(r3531025);
        double r3531027 = r3531026 - r3531005;
        double r3531028 = r3531020 * r3531027;
        double r3531029 = -1.0;
        double r3531030 = r3531009 * r3531029;
        double r3531031 = r3531016 ? r3531028 : r3531030;
        double r3531032 = r3531007 ? r3531014 : r3531031;
        return r3531032;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.0
Herbie	9.6

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

Derivation

1. Split input into 3 regimes

2. if b < -3.450829996567048e+138

   1. Initial program 58.5

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

   2. Simplified 58.5

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

   4. Applied add-cube-cbrt 58.5

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

   5. Applied associate-*r* 58.5

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

   7. Applied *-un-lft-identity 58.5

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

   8. Applied *-un-lft-identity 58.5

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

   9. Applied distribute-lft-out-- 58.5

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

   10. Simplified 58.5

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

   11. Taylor expanded around -inf 2.0

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

   12. Simplified 2.0

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

   if -3.450829996567048e+138 < b < 4.626043257219638e-62

   1. Initial program 12.3

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

   2. Simplified 12.3

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

   4. Applied add-cube-cbrt 12.6

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

   5. Applied associate-*r* 12.6

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

   7. Applied *-un-lft-identity 12.6

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

   8. Applied *-un-lft-identity 12.6

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

   9. Applied distribute-lft-out-- 12.6

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

   10. Simplified 12.3

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

   12. Applied *-un-lft-identity 12.3

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

   13. Applied div-inv 12.3

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

   14. Applied times-frac 12.4

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

   15. Simplified 12.4

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

   if 4.626043257219638e-62 < b

   1. Initial program 53.7

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

   2. Simplified 53.7

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

   4. Applied add-cube-cbrt 53.8

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

   5. Applied associate-*r* 53.8

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

   7. Applied *-un-lft-identity 53.8

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

   8. Applied *-un-lft-identity 53.8

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

   9. Applied distribute-lft-out-- 53.8

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

   10. Simplified 53.7

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

   11. Taylor expanded around inf 8.5

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))