NMSE problem 3.2.1

Average Error: 34.1 → 9.3

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.087668654677018032633364446323411964642 \cdot 10^{130}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r4357060 = b_2;
        double r4357061 = -r4357060;
        double r4357062 = r4357060 * r4357060;
        double r4357063 = a;
        double r4357064 = c;
        double r4357065 = r4357063 * r4357064;
        double r4357066 = r4357062 - r4357065;
        double r4357067 = sqrt(r4357066);
        double r4357068 = r4357061 - r4357067;
        double r4357069 = r4357068 / r4357063;
        return r4357069;
}

↓

double f(double a, double b_2, double c) {
        double r4357070 = b_2;
        double r4357071 = -4.356959927988237e-56;
        bool r4357072 = r4357070 <= r4357071;
        double r4357073 = -0.5;
        double r4357074 = c;
        double r4357075 = r4357074 / r4357070;
        double r4357076 = r4357073 * r4357075;
        double r4357077 = 3.087668654677018e+130;
        bool r4357078 = r4357070 <= r4357077;
        double r4357079 = -r4357070;
        double r4357080 = r4357070 * r4357070;
        double r4357081 = a;
        double r4357082 = r4357081 * r4357074;
        double r4357083 = r4357080 - r4357082;
        double r4357084 = sqrt(r4357083);
        double r4357085 = r4357079 - r4357084;
        double r4357086 = r4357085 / r4357081;
        double r4357087 = 0.5;
        double r4357088 = -2.0;
        double r4357089 = r4357070 / r4357081;
        double r4357090 = r4357088 * r4357089;
        double r4357091 = fma(r4357087, r4357075, r4357090);
        double r4357092 = r4357078 ? r4357086 : r4357091;
        double r4357093 = r4357072 ? r4357076 : r4357092;
        return r4357093;
}

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 < -4.356959927988237e-56

   1. Initial program 53.9

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

   2. Taylor expanded around -inf 7.7

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

   if -4.356959927988237e-56 < b_2 < 3.087668654677018e+130

   1. Initial program 12.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   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 12.5

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

   if 3.087668654677018e+130 < b_2

   1. Initial program 56.1

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

   2. Taylor expanded around inf 2.4

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

   3. Simplified 2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.087668654677018032633364446323411964642 \cdot 10^{130}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))