The quadratic formula (r2)

Average Error: 32.8 → 10.2

Time: 29.4s

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 r1706013 = b;
        double r1706014 = -r1706013;
        double r1706015 = r1706013 * r1706013;
        double r1706016 = 4.0;
        double r1706017 = a;
        double r1706018 = c;
        double r1706019 = r1706017 * r1706018;
        double r1706020 = r1706016 * r1706019;
        double r1706021 = r1706015 - r1706020;
        double r1706022 = sqrt(r1706021);
        double r1706023 = r1706014 - r1706022;
        double r1706024 = 2.0;
        double r1706025 = r1706024 * r1706017;
        double r1706026 = r1706023 / r1706025;
        return r1706026;
}

↓

double f(double a, double b, double c) {
        double r1706027 = b;
        double r1706028 = -3.031575300615258e-39;
        bool r1706029 = r1706027 <= r1706028;
        double r1706030 = c;
        double r1706031 = r1706030 / r1706027;
        double r1706032 = -r1706031;
        double r1706033 = 3.2912625180676585e+122;
        bool r1706034 = r1706027 <= r1706033;
        double r1706035 = -r1706027;
        double r1706036 = a;
        double r1706037 = r1706030 * r1706036;
        double r1706038 = -4.0;
        double r1706039 = r1706027 * r1706027;
        double r1706040 = fma(r1706037, r1706038, r1706039);
        double r1706041 = sqrt(r1706040);
        double r1706042 = r1706035 - r1706041;
        double r1706043 = 0.5;
        double r1706044 = r1706036 / r1706043;
        double r1706045 = r1706042 / r1706044;
        double r1706046 = r1706027 / r1706036;
        double r1706047 = r1706031 - r1706046;
        double r1706048 = r1706034 ? r1706045 : r1706047;
        double r1706049 = r1706029 ? r1706032 : r1706048;
        return r1706049;
}

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 "The quadratic formula (r2)"

  :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)))