quadm (p42, negative)

Average Error: 33.3 → 6.5

Time: 44.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}\\ \mathbf{elif}\;b \le -1.1213491597431702 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 3.843928652480689 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r4387119 = b;
        double r4387120 = -r4387119;
        double r4387121 = r4387119 * r4387119;
        double r4387122 = 4.0;
        double r4387123 = a;
        double r4387124 = c;
        double r4387125 = r4387123 * r4387124;
        double r4387126 = r4387122 * r4387125;
        double r4387127 = r4387121 - r4387126;
        double r4387128 = sqrt(r4387127);
        double r4387129 = r4387120 - r4387128;
        double r4387130 = 2.0;
        double r4387131 = r4387130 * r4387123;
        double r4387132 = r4387129 / r4387131;
        return r4387132;
}

↓

double f(double a, double b, double c) {
        double r4387133 = b;
        double r4387134 = -3.303853124735619e+50;
        bool r4387135 = r4387133 <= r4387134;
        double r4387136 = 2.0;
        double r4387137 = c;
        double r4387138 = r4387136 * r4387137;
        double r4387139 = r4387137 / r4387133;
        double r4387140 = a;
        double r4387141 = r4387139 * r4387140;
        double r4387142 = r4387141 - r4387133;
        double r4387143 = r4387136 * r4387142;
        double r4387144 = r4387138 / r4387143;
        double r4387145 = -1.1213491597431702e-288;
        bool r4387146 = r4387133 <= r4387145;
        double r4387147 = -r4387133;
        double r4387148 = r4387133 * r4387133;
        double r4387149 = 4.0;
        double r4387150 = r4387140 * r4387137;
        double r4387151 = r4387149 * r4387150;
        double r4387152 = r4387148 - r4387151;
        double r4387153 = sqrt(r4387152);
        double r4387154 = r4387147 + r4387153;
        double r4387155 = r4387138 / r4387154;
        double r4387156 = 3.843928652480689e+118;
        bool r4387157 = r4387133 <= r4387156;
        double r4387158 = 1.0;
        double r4387159 = r4387136 * r4387140;
        double r4387160 = r4387158 / r4387159;
        double r4387161 = r4387147 - r4387153;
        double r4387162 = r4387160 * r4387161;
        double r4387163 = r4387147 / r4387140;
        double r4387164 = r4387157 ? r4387162 : r4387163;
        double r4387165 = r4387146 ? r4387155 : r4387164;
        double r4387166 = r4387135 ? r4387144 : r4387165;
        return r4387166;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.2
Herbie	6.5

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

Derivation

1. Split input into 4 regimes

2. if b < -3.303853124735619e+50

   1. Initial program 55.7

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

   3. Applied div-inv 55.7

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

   5. Applied flip-- 55.8

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

   6. Applied associate-*l/ 55.8

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

   7. Simplified 26.1

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

   8. Taylor expanded around inf 24.3

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

   9. Taylor expanded around -inf 6.8

      \[\leadsto \frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\]

   10. Simplified 3.7

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

   if -3.303853124735619e+50 < b < -1.1213491597431702e-288

   1. Initial program 29.8

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

   3. Applied div-inv 29.9

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

   5. Applied flip-- 30.0

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

   6. Applied associate-*l/ 30.0

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

   7. Simplified 16.5

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

   8. Taylor expanded around inf 8.5

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

   if -1.1213491597431702e-288 < b < 3.843928652480689e+118

   1. Initial program 8.5

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

   3. Applied div-inv 8.6

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

   if 3.843928652480689e+118 < b

   1. Initial program 50.9

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

   3. Applied div-inv 50.9

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

   5. Applied flip-- 61.9

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

   6. Applied associate-*l/ 61.9

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

   7. Simplified 62.1

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

   8. Taylor expanded around inf 62.0

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

   9. Taylor expanded around 0 3.7

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

   10. Simplified 3.7

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}\\ \mathbf{elif}\;b \le -1.1213491597431702 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 3.843928652480689 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))