The quadratic formula (r2)

Average Error: 34.5 → 10.6

Time: 12.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r76863 = b;
        double r76864 = -r76863;
        double r76865 = r76863 * r76863;
        double r76866 = 4.0;
        double r76867 = a;
        double r76868 = c;
        double r76869 = r76867 * r76868;
        double r76870 = r76866 * r76869;
        double r76871 = r76865 - r76870;
        double r76872 = sqrt(r76871);
        double r76873 = r76864 - r76872;
        double r76874 = 2.0;
        double r76875 = r76874 * r76867;
        double r76876 = r76873 / r76875;
        return r76876;
}

↓

double f(double a, double b, double c) {
        double r76877 = b;
        double r76878 = -4.7828589349284326e-126;
        bool r76879 = r76877 <= r76878;
        double r76880 = -1.0;
        double r76881 = c;
        double r76882 = r76881 / r76877;
        double r76883 = r76880 * r76882;
        double r76884 = 3.6627135292415903e+111;
        bool r76885 = r76877 <= r76884;
        double r76886 = -r76877;
        double r76887 = r76877 * r76877;
        double r76888 = 4.0;
        double r76889 = a;
        double r76890 = r76889 * r76881;
        double r76891 = r76888 * r76890;
        double r76892 = r76887 - r76891;
        double r76893 = sqrt(r76892);
        double r76894 = r76886 - r76893;
        double r76895 = 1.0;
        double r76896 = 2.0;
        double r76897 = r76896 * r76889;
        double r76898 = r76895 / r76897;
        double r76899 = r76894 * r76898;
        double r76900 = -2.0;
        double r76901 = r76900 * r76877;
        double r76902 = r76901 / r76897;
        double r76903 = r76885 ? r76899 : r76902;
        double r76904 = r76879 ? r76883 : r76903;
        return r76904;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.5
Target	20.6
Herbie	10.6

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

Derivation

1. Split input into 3 regimes

2. if b < -4.7828589349284326e-126

   1. Initial program 51.3

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

   2. Taylor expanded around -inf 11.3

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

   if -4.7828589349284326e-126 < b < 3.6627135292415903e+111

   1. Initial program 12.0

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

   3. Applied div-inv 12.1

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

   if 3.6627135292415903e+111 < b

   1. Initial program 49.7

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

   3. Applied flip-- 63.3

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

   4. Simplified 62.3

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

   5. Simplified 62.3

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

   7. Applied add-sqr-sqrt 62.4

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

   8. Applied associate-/r* 62.4

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

   9. Simplified 62.4

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

   10. Taylor expanded around 0 3.7

       \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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