NMSE problem 3.2.1

Average Error: 34.5 → 10.1

Time: 4.9s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r113096 = b_2;
        double r113097 = -r113096;
        double r113098 = r113096 * r113096;
        double r113099 = a;
        double r113100 = c;
        double r113101 = r113099 * r113100;
        double r113102 = r113098 - r113101;
        double r113103 = sqrt(r113102);
        double r113104 = r113097 - r113103;
        double r113105 = r113104 / r113099;
        return r113105;
}

↓

double f(double a, double b_2, double c) {
        double r113106 = b_2;
        double r113107 = -4.706781135059312e-92;
        bool r113108 = r113106 <= r113107;
        double r113109 = -0.5;
        double r113110 = c;
        double r113111 = r113110 / r113106;
        double r113112 = r113109 * r113111;
        double r113113 = 5.722235152988638e+98;
        bool r113114 = r113106 <= r113113;
        double r113115 = -r113106;
        double r113116 = r113106 * r113106;
        double r113117 = a;
        double r113118 = r113117 * r113110;
        double r113119 = r113116 - r113118;
        double r113120 = sqrt(r113119);
        double r113121 = r113115 - r113120;
        double r113122 = r113121 / r113117;
        double r113123 = 0.5;
        double r113124 = r113123 * r113111;
        double r113125 = 2.0;
        double r113126 = r113106 / r113117;
        double r113127 = r113125 * r113126;
        double r113128 = r113124 - r113127;
        double r113129 = r113114 ? r113122 : r113128;
        double r113130 = r113108 ? r113112 : r113129;
        return r113130;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.706781135059312e-92

   1. Initial program 52.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 10.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -4.706781135059312e-92 < b_2 < 5.722235152988638e+98

   1. Initial program 12.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-inv 12.7

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
   4. Using strategy rm

   5. Applied un-div-inv 12.6

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

   if 5.722235152988638e+98 < b_2

   1. Initial program 47.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 3.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))