The quadratic formula (r1)

Average Error: 33.3 → 6.4

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r20365603 = b;
        double r20365604 = -r20365603;
        double r20365605 = r20365603 * r20365603;
        double r20365606 = 4.0;
        double r20365607 = a;
        double r20365608 = r20365606 * r20365607;
        double r20365609 = c;
        double r20365610 = r20365608 * r20365609;
        double r20365611 = r20365605 - r20365610;
        double r20365612 = sqrt(r20365611);
        double r20365613 = r20365604 + r20365612;
        double r20365614 = 2.0;
        double r20365615 = r20365614 * r20365607;
        double r20365616 = r20365613 / r20365615;
        return r20365616;
}

↓

double f(double a, double b, double c) {
        double r20365617 = b;
        double r20365618 = -3.263941314600607e+152;
        bool r20365619 = r20365617 <= r20365618;
        double r20365620 = c;
        double r20365621 = r20365620 / r20365617;
        double r20365622 = a;
        double r20365623 = r20365617 / r20365622;
        double r20365624 = r20365621 - r20365623;
        double r20365625 = -4.687918346756617e-254;
        bool r20365626 = r20365617 <= r20365625;
        double r20365627 = r20365617 * r20365617;
        double r20365628 = 4.0;
        double r20365629 = r20365620 * r20365622;
        double r20365630 = r20365628 * r20365629;
        double r20365631 = r20365627 - r20365630;
        double r20365632 = sqrt(r20365631);
        double r20365633 = r20365632 - r20365617;
        double r20365634 = r20365633 / r20365622;
        double r20365635 = 0.5;
        double r20365636 = r20365634 * r20365635;
        double r20365637 = 3.463606471108268e+121;
        bool r20365638 = r20365617 <= r20365637;
        double r20365639 = -2.0;
        double r20365640 = r20365620 * r20365639;
        double r20365641 = r20365632 + r20365617;
        double r20365642 = r20365640 / r20365641;
        double r20365643 = -r20365621;
        double r20365644 = r20365638 ? r20365642 : r20365643;
        double r20365645 = r20365626 ? r20365636 : r20365644;
        double r20365646 = r20365619 ? r20365624 : r20365645;
        return r20365646;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.3
Herbie	6.4

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

Derivation

1. Split input into 4 regimes

2. if b < -3.263941314600607e+152

   1. Initial program 60.1

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

   2. Simplified 60.1

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

   3. Taylor expanded around -inf 2.3

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

   if -3.263941314600607e+152 < b < -4.687918346756617e-254

   1. Initial program 7.8

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

   2. Simplified 7.8

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

   4. Applied clear-num 8.0

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

   6. Applied *-un-lft-identity 8.0

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

   7. Applied *-un-lft-identity 8.0

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

   8. Applied distribute-lft-out-- 8.0

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

   9. Applied times-frac 8.0

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

   10. Applied *-un-lft-identity 8.0

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

   11. Applied times-frac 8.0

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

   12. Simplified 8.0

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

   13. Simplified 7.8

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

   if -4.687918346756617e-254 < b < 3.463606471108268e+121

   1. Initial program 31.6

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

   2. Simplified 31.6

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

   4. Applied clear-num 31.7

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

   6. Applied flip-- 31.8

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

   7. Applied associate-/r/ 31.9

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

   8. Applied associate-/r* 31.9

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

   9. Simplified 14.4

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

   10. Taylor expanded around 0 8.7

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

   if 3.463606471108268e+121 < b

   1. Initial program 59.8

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

   2. Simplified 59.8

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

   4. Applied div-inv 59.8

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

   5. Simplified 59.8

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

   6. Taylor expanded around inf 2.3

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

   7. Simplified 2.3

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))