NMSE problem 3.2.1

Average Error: 34.5 → 10.6

Time: 11.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r79052 = b_2;
        double r79053 = -r79052;
        double r79054 = r79052 * r79052;
        double r79055 = a;
        double r79056 = c;
        double r79057 = r79055 * r79056;
        double r79058 = r79054 - r79057;
        double r79059 = sqrt(r79058);
        double r79060 = r79053 - r79059;
        double r79061 = r79060 / r79055;
        return r79061;
}

↓

double f(double a, double b_2, double c) {
        double r79062 = b_2;
        double r79063 = -4.7828589349284326e-126;
        bool r79064 = r79062 <= r79063;
        double r79065 = -0.5;
        double r79066 = c;
        double r79067 = r79066 / r79062;
        double r79068 = r79065 * r79067;
        double r79069 = 3.6627135292415903e+111;
        bool r79070 = r79062 <= r79069;
        double r79071 = -r79062;
        double r79072 = r79062 * r79062;
        double r79073 = a;
        double r79074 = r79073 * r79066;
        double r79075 = r79072 - r79074;
        double r79076 = sqrt(r79075);
        double r79077 = r79071 - r79076;
        double r79078 = 1.0;
        double r79079 = r79078 / r79073;
        double r79080 = r79077 * r79079;
        double r79081 = -2.0;
        double r79082 = r79081 * r79062;
        double r79083 = r79082 / r79073;
        double r79084 = r79070 ? r79080 : r79083;
        double r79085 = r79064 ? r79068 : r79084;
        return r79085;
}

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 < 3.6627135292415903e+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 3.6627135292415903e+111 < b_2

   1. Initial program 49.7

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

   3. Applied flip-- 63.3

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

   4. Simplified 62.3

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 62.3

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

   6. Taylor expanded around 0 3.6

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

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