Average Error: 33.2 → 9.8
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b}{a} \cdot \frac{-1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r1570301 = b;
        double r1570302 = -r1570301;
        double r1570303 = r1570301 * r1570301;
        double r1570304 = 4.0;
        double r1570305 = a;
        double r1570306 = c;
        double r1570307 = r1570305 * r1570306;
        double r1570308 = r1570304 * r1570307;
        double r1570309 = r1570303 - r1570308;
        double r1570310 = sqrt(r1570309);
        double r1570311 = r1570302 - r1570310;
        double r1570312 = 2.0;
        double r1570313 = r1570312 * r1570305;
        double r1570314 = r1570311 / r1570313;
        return r1570314;
}
↓
double f(double a, double b, double c) {
        double r1570315 = b;
        double r1570316 = -1.8774910265390396e-73;
        bool r1570317 = r1570315 <= r1570316;
        double r1570318 = c;
        double r1570319 = r1570318 / r1570315;
        double r1570320 = -r1570319;
        double r1570321 = 2.5703497435733685e+102;
        bool r1570322 = r1570315 <= r1570321;
        double r1570323 = r1570315 * r1570315;
        double r1570324 = 4.0;
        double r1570325 = r1570318 * r1570324;
        double r1570326 = a;
        double r1570327 = r1570325 * r1570326;
        double r1570328 = r1570323 - r1570327;
        double r1570329 = sqrt(r1570328);
        double r1570330 = r1570329 + r1570315;
        double r1570331 = r1570330 / r1570326;
        double r1570332 = -0.5;
        double r1570333 = r1570331 * r1570332;
        double r1570334 = r1570315 / r1570326;
        double r1570335 = r1570319 - r1570334;
        double r1570336 = r1570322 ? r1570333 : r1570335;
        double r1570337 = r1570317 ? r1570320 : r1570336;
        return r1570337;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.2
Target20.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8774910265390396e-73

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 52.5

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

    3. Simplified52.5

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

    4. Taylor expanded around -inf 8.6

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

    5. Simplified8.6

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

    if -1.8774910265390396e-73 < b < 2.5703497435733685e+102

    1. Initial program 13.1

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

    2. Taylor expanded around -inf 13.1

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

    3. Simplified13.1

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

    5. Applied *-un-lft-identity13.1

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

    6. Applied *-un-lft-identity13.1

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

    7. Applied distribute-rgt-neg-in13.1

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

    8. Applied distribute-lft-out--13.1

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

    9. Applied associate-/l*13.2

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

    11. Applied *-un-lft-identity13.2

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

    12. Applied times-frac13.2

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

    13. Applied add-sqr-sqrt13.2

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

    14. Applied times-frac13.2

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

    15. Simplified13.2

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

    16. Simplified13.1

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

    if 2.5703497435733685e+102 < b

    1. Initial program 43.9

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b}{a} \cdot \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))