The quadratic formula (r1)

Average Error: 34.7 → 8.9

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.4369139762720996 \cdot 10^{56}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.70350838245532258 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.8597470564587674 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r155067 = b;
        double r155068 = -r155067;
        double r155069 = r155067 * r155067;
        double r155070 = 4.0;
        double r155071 = a;
        double r155072 = r155070 * r155071;
        double r155073 = c;
        double r155074 = r155072 * r155073;
        double r155075 = r155069 - r155074;
        double r155076 = sqrt(r155075);
        double r155077 = r155068 + r155076;
        double r155078 = 2.0;
        double r155079 = r155078 * r155071;
        double r155080 = r155077 / r155079;
        return r155080;
}

↓

double f(double a, double b, double c) {
        double r155081 = b;
        double r155082 = -5.4369139762720996e+56;
        bool r155083 = r155081 <= r155082;
        double r155084 = 1.0;
        double r155085 = c;
        double r155086 = r155085 / r155081;
        double r155087 = a;
        double r155088 = r155081 / r155087;
        double r155089 = r155086 - r155088;
        double r155090 = r155084 * r155089;
        double r155091 = -8.703508382455323e-221;
        bool r155092 = r155081 <= r155091;
        double r155093 = 4.0;
        double r155094 = 2.0;
        double r155095 = pow(r155081, r155094);
        double r155096 = r155095 - r155095;
        double r155097 = r155087 * r155085;
        double r155098 = r155093 * r155097;
        double r155099 = r155096 + r155098;
        double r155100 = r155099 / r155097;
        double r155101 = r155093 / r155100;
        double r155102 = -r155081;
        double r155103 = r155081 * r155081;
        double r155104 = r155093 * r155087;
        double r155105 = r155104 * r155085;
        double r155106 = r155103 - r155105;
        double r155107 = sqrt(r155106);
        double r155108 = r155102 + r155107;
        double r155109 = r155101 * r155108;
        double r155110 = 2.0;
        double r155111 = r155110 * r155087;
        double r155112 = r155109 / r155111;
        double r155113 = 1.8597470564587674e+138;
        bool r155114 = r155081 <= r155113;
        double r155115 = 1.0;
        double r155116 = r155102 - r155107;
        double r155117 = r155115 / r155116;
        double r155118 = r155110 / r155093;
        double r155119 = r155118 * r155115;
        double r155120 = r155119 / r155085;
        double r155121 = r155117 / r155120;
        double r155122 = r155115 * r155121;
        double r155123 = -1.0;
        double r155124 = r155123 * r155086;
        double r155125 = r155114 ? r155122 : r155124;
        double r155126 = r155092 ? r155112 : r155125;
        double r155127 = r155083 ? r155090 : r155126;
        return r155127;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.7
Target	21.9
Herbie	8.9

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

Derivation

1. Split input into 4 regimes

2. if b < -5.4369139762720996e+56

   1. Initial program 42.7

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

   2. Taylor expanded around -inf 5.1

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

   3. Simplified 5.1

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

   if -5.4369139762720996e+56 < b < -8.703508382455323e-221

   1. Initial program 8.3

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

   3. Applied flip-+ 35.2

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

   4. Simplified 35.3

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

   6. Applied flip-- 35.3

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

   7. Applied associate-/r/ 35.4

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

   8. Simplified 16.9

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

   if -8.703508382455323e-221 < b < 1.8597470564587674e+138

   1. Initial program 31.1

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

   3. Applied flip-+ 31.2

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

   4. Simplified 16.1

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

   6. Applied *-un-lft-identity 16.1

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

   7. Applied *-un-lft-identity 16.1

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

   8. Applied times-frac 16.1

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

   9. Applied associate-/l* 16.2

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

   10. Simplified 15.0

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

   12. Applied div-inv 15.0

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

   13. Simplified 9.8

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

   if 1.8597470564587674e+138 < b

   1. Initial program 62.4

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

   2. Taylor expanded around inf 2.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4369139762720996 \cdot 10^{56}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.70350838245532258 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.8597470564587674 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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