The quadratic formula (r2)

Average Error: 33.8 → 6.7

Time: 34.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3422084503380959 \cdot 10^{+126}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.9582179666484207 \cdot 10^{-229}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 9.112814637305151 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3789764 = b;
        double r3789765 = -r3789764;
        double r3789766 = r3789764 * r3789764;
        double r3789767 = 4.0;
        double r3789768 = a;
        double r3789769 = c;
        double r3789770 = r3789768 * r3789769;
        double r3789771 = r3789767 * r3789770;
        double r3789772 = r3789766 - r3789771;
        double r3789773 = sqrt(r3789772);
        double r3789774 = r3789765 - r3789773;
        double r3789775 = 2.0;
        double r3789776 = r3789775 * r3789768;
        double r3789777 = r3789774 / r3789776;
        return r3789777;
}

↓

double f(double a, double b, double c) {
        double r3789778 = b;
        double r3789779 = -1.3422084503380959e+126;
        bool r3789780 = r3789778 <= r3789779;
        double r3789781 = c;
        double r3789782 = -r3789781;
        double r3789783 = r3789782 / r3789778;
        double r3789784 = -2.9582179666484207e-229;
        bool r3789785 = r3789778 <= r3789784;
        double r3789786 = 2.0;
        double r3789787 = r3789786 * r3789781;
        double r3789788 = r3789778 * r3789778;
        double r3789789 = 4.0;
        double r3789790 = a;
        double r3789791 = r3789790 * r3789781;
        double r3789792 = r3789789 * r3789791;
        double r3789793 = r3789788 - r3789792;
        double r3789794 = sqrt(r3789793);
        double r3789795 = r3789794 - r3789778;
        double r3789796 = sqrt(r3789795);
        double r3789797 = r3789787 / r3789796;
        double r3789798 = 1.0;
        double r3789799 = r3789798 / r3789796;
        double r3789800 = r3789797 * r3789799;
        double r3789801 = 9.112814637305151e+83;
        bool r3789802 = r3789778 <= r3789801;
        double r3789803 = 0.5;
        double r3789804 = r3789803 / r3789790;
        double r3789805 = -r3789778;
        double r3789806 = r3789805 - r3789794;
        double r3789807 = r3789804 * r3789806;
        double r3789808 = r3789781 / r3789778;
        double r3789809 = r3789778 / r3789790;
        double r3789810 = r3789808 - r3789809;
        double r3789811 = r3789802 ? r3789807 : r3789810;
        double r3789812 = r3789785 ? r3789800 : r3789811;
        double r3789813 = r3789780 ? r3789783 : r3789812;
        return r3789813;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.8
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -1.3422084503380959e+126

   1. Initial program 59.9

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

   2. Taylor expanded around -inf 2.0

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

   3. Simplified 2.0

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

   if -1.3422084503380959e+126 < b < -2.9582179666484207e-229

   1. Initial program 37.3

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

   3. Applied clear-num 37.3

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

   5. Applied flip-- 37.4

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

   6. Applied associate-/r/ 37.5

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

   7. Applied add-sqr-sqrt 37.5

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

   8. Applied times-frac 37.5

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

   9. Simplified 15.9

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot \left(0 - \left(-4 \cdot c\right) \cdot a\right)\right)} \cdot \frac{\sqrt{1}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   10. Simplified 15.9

       \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot \left(0 - \left(-4 \cdot c\right) \cdot a\right)\right) \cdot \color{blue}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   11. Taylor expanded around inf 7.4

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

   13. Applied add-sqr-sqrt 7.6

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

   14. Applied add-cube-cbrt 7.6

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

   15. Applied times-frac 7.7

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

   16. Applied associate-*r* 7.7

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

   17. Simplified 7.6

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

   if -2.9582179666484207e-229 < b < 9.112814637305151e+83

   1. Initial program 10.0

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

   3. Applied div-inv 10.1

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

   4. Simplified 10.1

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

   if 9.112814637305151e+83 < b

   1. Initial program 42.5

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

   2. Taylor expanded around inf 4.1

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3422084503380959 \cdot 10^{+126}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.9582179666484207 \cdot 10^{-229}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 9.112814637305151 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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)))