quadm (p42, negative)

Average Error: 32.6 → 9.7

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1796228 = b;
        double r1796229 = -r1796228;
        double r1796230 = r1796228 * r1796228;
        double r1796231 = 4.0;
        double r1796232 = a;
        double r1796233 = c;
        double r1796234 = r1796232 * r1796233;
        double r1796235 = r1796231 * r1796234;
        double r1796236 = r1796230 - r1796235;
        double r1796237 = sqrt(r1796236);
        double r1796238 = r1796229 - r1796237;
        double r1796239 = 2.0;
        double r1796240 = r1796239 * r1796232;
        double r1796241 = r1796238 / r1796240;
        return r1796241;
}

↓

double f(double a, double b, double c) {
        double r1796242 = b;
        double r1796243 = -6.90131991727783e-39;
        bool r1796244 = r1796242 <= r1796243;
        double r1796245 = c;
        double r1796246 = r1796245 / r1796242;
        double r1796247 = -r1796246;
        double r1796248 = 4.012768074517757e+87;
        bool r1796249 = r1796242 <= r1796248;
        double r1796250 = 0.5;
        double r1796251 = a;
        double r1796252 = r1796250 / r1796251;
        double r1796253 = 1.0;
        double r1796254 = -r1796242;
        double r1796255 = r1796242 * r1796242;
        double r1796256 = r1796245 * r1796251;
        double r1796257 = 4.0;
        double r1796258 = r1796256 * r1796257;
        double r1796259 = r1796255 - r1796258;
        double r1796260 = sqrt(r1796259);
        double r1796261 = r1796254 - r1796260;
        double r1796262 = r1796253 / r1796261;
        double r1796263 = r1796252 / r1796262;
        double r1796264 = r1796254 / r1796251;
        double r1796265 = r1796249 ? r1796263 : r1796264;
        double r1796266 = r1796244 ? r1796247 : r1796265;
        return r1796266;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.6
Target	20.0
Herbie	9.7

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

Derivation

1. Split input into 3 regimes

2. if b < -6.90131991727783e-39

   1. Initial program 53.0

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

   2. Taylor expanded around -inf 7.9

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

   3. Simplified 7.9

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

   if -6.90131991727783e-39 < b < 4.012768074517757e+87

   1. Initial program 13.2

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

   3. Applied *-un-lft-identity 13.2

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

   4. Applied associate-/l* 13.4

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

   6. Applied div-inv 13.4

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

   7. Applied associate-/r* 13.4

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

   8. Simplified 13.4

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

   if 4.012768074517757e+87 < b

   1. Initial program 41.3

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

   3. Applied *-un-lft-identity 41.3

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

   4. Applied associate-/l* 41.3

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

   6. Applied div-inv 41.4

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

   7. Applied associate-/r* 41.4

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

   8. Simplified 41.4

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

   9. Taylor expanded around 0 3.3

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

   10. Simplified 3.3

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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)))