quadm (p42, negative)

Average Error: 32.8 → 6.4

Time: 3.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.1214768270116103 \cdot 10^{+154}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.199441090208904 \cdot 10^{-250}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}\\ \mathbf{elif}\;b \le 3.3389954009657566 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r17923813 = b;
        double r17923814 = -r17923813;
        double r17923815 = r17923813 * r17923813;
        double r17923816 = 4.0;
        double r17923817 = a;
        double r17923818 = c;
        double r17923819 = r17923817 * r17923818;
        double r17923820 = r17923816 * r17923819;
        double r17923821 = r17923815 - r17923820;
        double r17923822 = sqrt(r17923821);
        double r17923823 = r17923814 - r17923822;
        double r17923824 = 2.0;
        double r17923825 = r17923824 * r17923817;
        double r17923826 = r17923823 / r17923825;
        return r17923826;
}

↓

double f(double a, double b, double c) {
        double r17923827 = b;
        double r17923828 = -1.1214768270116103e+154;
        bool r17923829 = r17923827 <= r17923828;
        double r17923830 = c;
        double r17923831 = r17923830 / r17923827;
        double r17923832 = -r17923831;
        double r17923833 = 1.199441090208904e-250;
        bool r17923834 = r17923827 <= r17923833;
        double r17923835 = a;
        double r17923836 = r17923830 * r17923835;
        double r17923837 = -4.0;
        double r17923838 = r17923827 * r17923827;
        double r17923839 = fma(r17923836, r17923837, r17923838);
        double r17923840 = sqrt(r17923839);
        double r17923841 = r17923840 - r17923827;
        double r17923842 = 2.0;
        double r17923843 = r17923841 / r17923842;
        double r17923844 = r17923830 / r17923843;
        double r17923845 = 3.3389954009657566e+124;
        bool r17923846 = r17923827 <= r17923845;
        double r17923847 = -r17923827;
        double r17923848 = fma(r17923837, r17923836, r17923838);
        double r17923849 = sqrt(r17923848);
        double r17923850 = r17923847 - r17923849;
        double r17923851 = r17923835 * r17923842;
        double r17923852 = r17923850 / r17923851;
        double r17923853 = r17923827 / r17923835;
        double r17923854 = -r17923853;
        double r17923855 = r17923846 ? r17923852 : r17923854;
        double r17923856 = r17923834 ? r17923844 : r17923855;
        double r17923857 = r17923829 ? r17923832 : r17923856;
        return r17923857;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.8
Target	20.1
Herbie	6.4

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

Derivation

1. Split input into 4 regimes

2. if b < -1.1214768270116103e+154

   1. Initial program 62.9

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

   2. Simplified 62.9

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

   3. Taylor expanded around -inf 1.5

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

   4. Simplified 1.5

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

   if -1.1214768270116103e+154 < b < 1.199441090208904e-250

   1. Initial program 32.2

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

   2. Simplified 32.2

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

   4. Applied flip-- 32.3

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

   5. Simplified 16.1

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

   6. Simplified 16.2

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

   8. Applied *-un-lft-identity 16.2

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

   9. Applied *-un-lft-identity 16.2

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

   10. Applied *-un-lft-identity 16.2

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

   11. Applied times-frac 16.2

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

   12. Applied times-frac 16.2

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

   13. Simplified 16.2

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

   14. Simplified 8.4

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

   if 1.199441090208904e-250 < b < 3.3389954009657566e+124

   1. Initial program 7.8

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

   2. Simplified 7.8

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

   4. Applied *-un-lft-identity 7.8

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

   5. Applied add-sqr-sqrt 8.8

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

   6. Applied *-un-lft-identity 8.8

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

   7. Applied times-frac 8.6

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

   8. Applied times-frac 8.5

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

   9. Simplified 8.5

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

   10. Simplified 7.8

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

   if 3.3389954009657566e+124 < b

   1. Initial program 50.5

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

   2. Simplified 50.5

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

   4. Applied flip-- 61.9

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

   5. Simplified 62.0

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

   6. Simplified 62.0

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

   7. Taylor expanded around 0 3.5

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

   8. Simplified 3.5

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1214768270116103 \cdot 10^{+154}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.199441090208904 \cdot 10^{-250}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}\\ \mathbf{elif}\;b \le 3.3389954009657566 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))