Average Error: 34.2 → 10.0
Time: 16.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;-\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a}}{2}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{-\left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\
\;\;\;\;\frac{-1 \cdot c}{b}\\

\mathbf{elif}\;b \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\
\;\;\;\;-\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a}}{2}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\

\mathbf{elif}\;b \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\
\;\;\;\;\frac{-1 \cdot c}{b}\\

\mathbf{elif}\;b \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{-\left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r52282 = b;
        double r52283 = -r52282;
        double r52284 = r52282 * r52282;
        double r52285 = 4.0;
        double r52286 = a;
        double r52287 = c;
        double r52288 = r52286 * r52287;
        double r52289 = r52285 * r52288;
        double r52290 = r52284 - r52289;
        double r52291 = sqrt(r52290);
        double r52292 = r52283 - r52291;
        double r52293 = 2.0;
        double r52294 = r52293 * r52286;
        double r52295 = r52292 / r52294;
        return r52295;
}
↓
double f(double a, double b, double c) {
        double r52296 = b;
        double r52297 = -5.84038254482515e+46;
        bool r52298 = r52296 <= r52297;
        double r52299 = -1.0;
        double r52300 = c;
        double r52301 = r52299 * r52300;
        double r52302 = r52301 / r52296;
        double r52303 = -7.877985662156599e-94;
        bool r52304 = r52296 <= r52303;
        double r52305 = r52296 * r52296;
        double r52306 = r52305 - r52305;
        double r52307 = a;
        double r52308 = 4.0;
        double r52309 = r52300 * r52308;
        double r52310 = r52307 * r52309;
        double r52311 = r52306 + r52310;
        double r52312 = r52311 / r52307;
        double r52313 = 2.0;
        double r52314 = r52312 / r52313;
        double r52315 = r52300 * r52307;
        double r52316 = r52308 * r52315;
        double r52317 = r52305 - r52316;
        double r52318 = sqrt(r52317);
        double r52319 = r52296 - r52318;
        double r52320 = r52314 / r52319;
        double r52321 = -r52320;
        double r52322 = -6.596302400897662e-136;
        bool r52323 = r52296 <= r52322;
        double r52324 = 7.501979458872916e+77;
        bool r52325 = r52296 <= r52324;
        double r52326 = 1.0;
        double r52327 = r52307 * r52313;
        double r52328 = r52318 + r52296;
        double r52329 = -r52328;
        double r52330 = r52327 / r52329;
        double r52331 = r52326 / r52330;
        double r52332 = r52300 / r52296;
        double r52333 = r52296 / r52307;
        double r52334 = r52332 - r52333;
        double r52335 = 1.0;
        double r52336 = r52334 * r52335;
        double r52337 = r52325 ? r52331 : r52336;
        double r52338 = r52323 ? r52302 : r52337;
        double r52339 = r52304 ? r52321 : r52338;
        double r52340 = r52298 ? r52302 : r52339;
        return r52340;
}

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

Original34.2
Target21.2
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.84038254482515e+46 or -7.877985662156599e-94 < b < -6.596302400897662e-136

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -5.84038254482515e+46 < b < -7.877985662156599e-94

    1. Initial program 40.2

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

    3. Applied clear-num40.2

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

    4. Simplified40.2

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

    6. Applied *-un-lft-identity40.2

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

    7. Applied add-cube-cbrt40.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{a \cdot 2}{-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]

    8. Applied times-frac40.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a \cdot 2}{-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}\]

    9. Simplified40.2

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a \cdot 2}{-\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]

    10. Simplified40.2

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

    12. Applied flip-+40.3

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

    13. Applied distribute-neg-frac40.3

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

    14. Applied associate-*r/40.3

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

    15. Simplified14.8

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

    if -6.596302400897662e-136 < b < 7.501979458872916e+77

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    4. Simplified12.2

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

    if 7.501979458872916e+77 < b

    1. Initial program 42.5

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

    2. Taylor expanded around inf 5.0

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

    3. Simplified5.0

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;-\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a}}{2}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{-\left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))