quadm (p42, negative)

Average Error: 32.8 → 10.2

Time: 32.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.2912625180676585 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1565919 = b;
        double r1565920 = -r1565919;
        double r1565921 = r1565919 * r1565919;
        double r1565922 = 4.0;
        double r1565923 = a;
        double r1565924 = c;
        double r1565925 = r1565923 * r1565924;
        double r1565926 = r1565922 * r1565925;
        double r1565927 = r1565921 - r1565926;
        double r1565928 = sqrt(r1565927);
        double r1565929 = r1565920 - r1565928;
        double r1565930 = 2.0;
        double r1565931 = r1565930 * r1565923;
        double r1565932 = r1565929 / r1565931;
        return r1565932;
}

↓

double f(double a, double b, double c) {
        double r1565933 = b;
        double r1565934 = -3.031575300615258e-39;
        bool r1565935 = r1565933 <= r1565934;
        double r1565936 = c;
        double r1565937 = r1565936 / r1565933;
        double r1565938 = -r1565937;
        double r1565939 = 3.2912625180676585e+122;
        bool r1565940 = r1565933 <= r1565939;
        double r1565941 = -r1565933;
        double r1565942 = a;
        double r1565943 = r1565936 * r1565942;
        double r1565944 = -4.0;
        double r1565945 = r1565933 * r1565933;
        double r1565946 = fma(r1565943, r1565944, r1565945);
        double r1565947 = sqrt(r1565946);
        double r1565948 = r1565941 - r1565947;
        double r1565949 = 0.5;
        double r1565950 = r1565942 / r1565949;
        double r1565951 = r1565948 / r1565950;
        double r1565952 = r1565933 / r1565942;
        double r1565953 = r1565937 - r1565952;
        double r1565954 = r1565940 ? r1565951 : r1565953;
        double r1565955 = r1565935 ? r1565938 : r1565954;
        return r1565955;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.8
Target	20.1
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if b < -3.031575300615258e-39

   1. Initial program 53.3

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

   2. Simplified 53.3

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

   3. Taylor expanded around -inf 7.8

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

   4. Simplified 7.8

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

   if -3.031575300615258e-39 < b < 3.2912625180676585e+122

   1. Initial program 13.8

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

   2. Simplified 13.8

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

   4. Applied *-un-lft-identity 13.8

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

   5. Applied associate-/l* 13.9

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

   7. Applied div-inv 13.9

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

   8. Applied *-un-lft-identity 13.9

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

   9. Applied times-frac 13.9

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

   10. Applied associate-/r* 13.8

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

   11. Simplified 13.8

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

   if 3.2912625180676585e+122 < b

   1. Initial program 49.9

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

   2. Simplified 49.9

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

   4. Applied *-un-lft-identity 49.9

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

   5. Applied associate-/l* 50.0

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

   7. Applied div-inv 50.0

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

   8. Applied *-un-lft-identity 50.0

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

   9. Applied times-frac 50.0

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

   10. Applied associate-/r* 50.0

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

   11. Simplified 49.9

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

   12. Taylor expanded around inf 3.3

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.2912625180676585 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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)))