NMSE problem 3.2.1

Average Error: 34.5 → 10.5

Time: 8.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r73015 = b_2;
        double r73016 = -r73015;
        double r73017 = r73015 * r73015;
        double r73018 = a;
        double r73019 = c;
        double r73020 = r73018 * r73019;
        double r73021 = r73017 - r73020;
        double r73022 = sqrt(r73021);
        double r73023 = r73016 - r73022;
        double r73024 = r73023 / r73018;
        return r73024;
}

↓

double f(double a, double b_2, double c) {
        double r73025 = b_2;
        double r73026 = -4.7828589349284326e-126;
        bool r73027 = r73025 <= r73026;
        double r73028 = -0.5;
        double r73029 = c;
        double r73030 = r73029 / r73025;
        double r73031 = r73028 * r73030;
        double r73032 = 3.1323104234385293e+111;
        bool r73033 = r73025 <= r73032;
        double r73034 = -r73025;
        double r73035 = r73025 * r73025;
        double r73036 = a;
        double r73037 = r73036 * r73029;
        double r73038 = r73035 - r73037;
        double r73039 = sqrt(r73038);
        double r73040 = r73034 - r73039;
        double r73041 = 1.0;
        double r73042 = r73041 / r73036;
        double r73043 = r73040 * r73042;
        double r73044 = 0.5;
        double r73045 = -2.0;
        double r73046 = r73025 / r73036;
        double r73047 = r73045 * r73046;
        double r73048 = fma(r73044, r73030, r73047);
        double r73049 = r73033 ? r73043 : r73048;
        double r73050 = r73027 ? r73031 : r73049;
        return r73050;
}

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 < -4.7828589349284326e-126

   1. Initial program 51.3

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

   2. Taylor expanded around -inf 11.3

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

   if -4.7828589349284326e-126 < b_2 < 3.1323104234385293e+111

   1. Initial program 12.0

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

   3. Applied div-inv 12.1

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

   if 3.1323104234385293e+111 < b_2

   1. Initial program 49.7

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

   2. Taylor expanded around inf 3.4

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

   3. Simplified 3.4

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))