quadm (p42, negative)

Average Error: 33.4 → 9.9

Time: 26.5s

Precision: 64

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1754154 = b;
        double r1754155 = -r1754154;
        double r1754156 = r1754154 * r1754154;
        double r1754157 = 4.0;
        double r1754158 = a;
        double r1754159 = c;
        double r1754160 = r1754158 * r1754159;
        double r1754161 = r1754157 * r1754160;
        double r1754162 = r1754156 - r1754161;
        double r1754163 = sqrt(r1754162);
        double r1754164 = r1754155 - r1754163;
        double r1754165 = 2.0;
        double r1754166 = r1754165 * r1754158;
        double r1754167 = r1754164 / r1754166;
        return r1754167;
}

↓

double f(double a, double b, double c) {
        double r1754168 = b;
        double r1754169 = -2.840085388791461e-68;
        bool r1754170 = r1754168 <= r1754169;
        double r1754171 = c;
        double r1754172 = r1754171 / r1754168;
        double r1754173 = -r1754172;
        double r1754174 = 1.5949594684703287e+126;
        bool r1754175 = r1754168 <= r1754174;
        double r1754176 = -r1754168;
        double r1754177 = -4.0;
        double r1754178 = a;
        double r1754179 = r1754178 * r1754171;
        double r1754180 = r1754168 * r1754168;
        double r1754181 = fma(r1754177, r1754179, r1754180);
        double r1754182 = sqrt(r1754181);
        double r1754183 = r1754176 - r1754182;
        double r1754184 = 0.5;
        double r1754185 = r1754183 * r1754184;
        double r1754186 = r1754185 / r1754178;
        double r1754187 = 2.0;
        double r1754188 = r1754168 / r1754178;
        double r1754189 = r1754171 / r1754188;
        double r1754190 = r1754189 - r1754168;
        double r1754191 = r1754187 * r1754190;
        double r1754192 = r1754191 / r1754187;
        double r1754193 = r1754192 / r1754178;
        double r1754194 = r1754175 ? r1754186 : r1754193;
        double r1754195 = r1754170 ? r1754173 : r1754194;
        return r1754195;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.4
Target	20.2
Herbie	9.9

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -2.840085388791461e-68

   1. Initial program 52.7

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 52.7

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 52.7

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{2}}{\color{blue}{1 \cdot a}}\]

   5. Applied div-inv 52.7

      \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}}{1 \cdot a}\]

   6. Applied times-frac 52.7

      \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{1} \cdot \frac{\frac{1}{2}}{a}}\]

   7. Simplified 52.7

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right)} \cdot \frac{\frac{1}{2}}{a}\]

   8. Simplified 52.7

      \[\leadsto \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
   9. Using strategy rm

   10. Applied associate-*r/ 52.7

       \[\leadsto \color{blue}{\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}{a}}\]

   11. Taylor expanded around -inf 9.0

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

   12. Simplified 9.0

       \[\leadsto \color{blue}{\frac{-c}{b}}\]

   if -2.840085388791461e-68 < b < 1.5949594684703287e+126

   1. Initial program 12.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 12.5

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 12.5

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{2}}{\color{blue}{1 \cdot a}}\]

   5. Applied div-inv 12.5

      \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}}{1 \cdot a}\]

   6. Applied times-frac 12.7

      \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{1} \cdot \frac{\frac{1}{2}}{a}}\]

   7. Simplified 12.7

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right)} \cdot \frac{\frac{1}{2}}{a}\]

   8. Simplified 12.7

      \[\leadsto \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
   9. Using strategy rm

   10. Applied associate-*r/ 12.5

       \[\leadsto \color{blue}{\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}{a}}\]

   if 1.5949594684703287e+126 < b

   1. Initial program 50.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 50.8

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}}\]

   3. Taylor expanded around inf 10.6

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

   4. Simplified 3.2

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))