Average Error: 34.2 → 8.1
Time: 6.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.55528137777049654 \cdot 10^{140}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.52089004516141949 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 26039420339585284:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.55528137777049654 \cdot 10^{140}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 9.52089004516141949 \cdot 10^{-271}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 26039420339585284:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r122283 = b;
        double r122284 = -r122283;
        double r122285 = r122283 * r122283;
        double r122286 = 4.0;
        double r122287 = a;
        double r122288 = r122286 * r122287;
        double r122289 = c;
        double r122290 = r122288 * r122289;
        double r122291 = r122285 - r122290;
        double r122292 = sqrt(r122291);
        double r122293 = r122284 + r122292;
        double r122294 = 2.0;
        double r122295 = r122294 * r122287;
        double r122296 = r122293 / r122295;
        return r122296;
}
↓
double f(double a, double b, double c) {
        double r122297 = b;
        double r122298 = -8.555281377770497e+140;
        bool r122299 = r122297 <= r122298;
        double r122300 = 1.0;
        double r122301 = c;
        double r122302 = r122301 / r122297;
        double r122303 = a;
        double r122304 = r122297 / r122303;
        double r122305 = r122302 - r122304;
        double r122306 = r122300 * r122305;
        double r122307 = 9.52089004516142e-271;
        bool r122308 = r122297 <= r122307;
        double r122309 = -r122297;
        double r122310 = r122297 * r122297;
        double r122311 = 4.0;
        double r122312 = r122311 * r122303;
        double r122313 = r122312 * r122301;
        double r122314 = r122310 - r122313;
        double r122315 = sqrt(r122314);
        double r122316 = r122309 + r122315;
        double r122317 = 2.0;
        double r122318 = r122317 * r122303;
        double r122319 = r122316 / r122318;
        double r122320 = 26039420339585284.0;
        bool r122321 = r122297 <= r122320;
        double r122322 = 1.0;
        double r122323 = r122309 - r122315;
        double r122324 = r122323 / r122311;
        double r122325 = r122324 / r122303;
        double r122326 = r122325 / r122301;
        double r122327 = r122322 / r122326;
        double r122328 = r122327 / r122318;
        double r122329 = -1.0;
        double r122330 = r122329 * r122302;
        double r122331 = r122321 ? r122328 : r122330;
        double r122332 = r122308 ? r122319 : r122331;
        double r122333 = r122299 ? r122306 : r122332;
        return r122333;
}

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
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.555281377770497e+140

    1. Initial program 58.5

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

    3. Applied flip-+63.9

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

    4. Simplified62.7

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num62.7

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{0 + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    7. Simplified62.7

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

    8. Taylor expanded around -inf 2.4

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

    9. Simplified2.4

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

    if -8.555281377770497e+140 < b < 9.52089004516142e-271

    1. Initial program 9.7

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

    if 9.52089004516142e-271 < b < 26039420339585284.0

    1. Initial program 27.1

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

    3. Applied flip-+27.2

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

    4. Simplified17.0

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num17.1

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{0 + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    7. Simplified17.1

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

    9. Applied associate-/r*14.2

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

    if 26039420339585284.0 < b

    1. Initial program 56.0

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

    3. Applied flip-+56.0

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

    4. Simplified26.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num26.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{0 + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    7. Simplified26.8

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

    8. Taylor expanded around inf 4.8

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.55528137777049654 \cdot 10^{140}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.52089004516141949 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 26039420339585284:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))