The quadratic formula (r1)

Average Error: 33.8 → 9.8

Time: 12.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r171159 = b;
        double r171160 = -r171159;
        double r171161 = r171159 * r171159;
        double r171162 = 4.0;
        double r171163 = a;
        double r171164 = r171162 * r171163;
        double r171165 = c;
        double r171166 = r171164 * r171165;
        double r171167 = r171161 - r171166;
        double r171168 = sqrt(r171167);
        double r171169 = r171160 + r171168;
        double r171170 = 2.0;
        double r171171 = r171170 * r171163;
        double r171172 = r171169 / r171171;
        return r171172;
}

↓

double f(double a, double b, double c) {
        double r171173 = b;
        double r171174 = -7.93152454634662e+153;
        bool r171175 = r171173 <= r171174;
        double r171176 = 1.0;
        double r171177 = c;
        double r171178 = r171177 / r171173;
        double r171179 = a;
        double r171180 = r171173 / r171179;
        double r171181 = r171178 - r171180;
        double r171182 = r171176 * r171181;
        double r171183 = 2.0569776426586135e-106;
        bool r171184 = r171173 <= r171183;
        double r171185 = 4.0;
        double r171186 = r171185 * r171179;
        double r171187 = r171177 * r171186;
        double r171188 = -r171187;
        double r171189 = fma(r171173, r171173, r171188);
        double r171190 = sqrt(r171189);
        double r171191 = r171190 - r171173;
        double r171192 = 1.0;
        double r171193 = 2.0;
        double r171194 = r171192 / r171193;
        double r171195 = r171179 / r171194;
        double r171196 = r171191 / r171195;
        double r171197 = -1.0;
        double r171198 = r171197 * r171178;
        double r171199 = r171184 ? r171196 : r171198;
        double r171200 = r171175 ? r171182 : r171199;
        return r171200;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	21.2
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -7.93152454634662e+153

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   3. Taylor expanded around -inf 1.9

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

   4. Simplified 1.9

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

   if -7.93152454634662e+153 < b < 2.0569776426586135e-106

   1. Initial program 11.2

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

   2. Simplified 11.2

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

   4. Applied clear-num 11.3

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

   6. Applied div-inv 11.3

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

   7. Applied *-un-lft-identity 11.3

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

   8. Applied times-frac 11.4

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

   9. Applied associate-/r* 11.3

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

   10. Simplified 11.2

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

   if 2.0569776426586135e-106 < b

   1. Initial program 52.0

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

   2. Simplified 52.0

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

   3. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

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