quad2m (problem 3.2.1, negative)

Average Error: 34.5 → 10.1

Time: 5.3s

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 r16036 = b_2;
        double r16037 = -r16036;
        double r16038 = r16036 * r16036;
        double r16039 = a;
        double r16040 = c;
        double r16041 = r16039 * r16040;
        double r16042 = r16038 - r16041;
        double r16043 = sqrt(r16042);
        double r16044 = r16037 - r16043;
        double r16045 = r16044 / r16039;
        return r16045;
}

↓

double f(double a, double b_2, double c) {
        double r16046 = b_2;
        double r16047 = -4.706781135059312e-92;
        bool r16048 = r16046 <= r16047;
        double r16049 = -0.5;
        double r16050 = c;
        double r16051 = r16050 / r16046;
        double r16052 = r16049 * r16051;
        double r16053 = 5.722235152988638e+98;
        bool r16054 = r16046 <= r16053;
        double r16055 = -r16046;
        double r16056 = r16046 * r16046;
        double r16057 = a;
        double r16058 = r16057 * r16050;
        double r16059 = r16056 - r16058;
        double r16060 = sqrt(r16059);
        double r16061 = r16055 - r16060;
        double r16062 = r16061 / r16057;
        double r16063 = 0.5;
        double r16064 = r16063 * r16051;
        double r16065 = 2.0;
        double r16066 = r16046 / r16057;
        double r16067 = r16065 * r16066;
        double r16068 = r16064 - r16067;
        double r16069 = r16054 ? r16062 : r16068;
        double r16070 = r16048 ? r16052 : r16069;
        return r16070;
}

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 "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))