Average Error: 34.0 → 8.5
Time: 13.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -977083.9042033920995891094207763671875:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.277624610006997683151417691157213452138 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 4.294531430978972552500215519031197589784 \cdot 10^{104}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -977083.9042033920995891094207763671875:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.277624610006997683151417691157213452138 \cdot 10^{-291}:\\
\;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 4.294531430978972552500215519031197589784 \cdot 10^{104}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r59409 = b;
        double r59410 = -r59409;
        double r59411 = r59409 * r59409;
        double r59412 = 4.0;
        double r59413 = a;
        double r59414 = c;
        double r59415 = r59413 * r59414;
        double r59416 = r59412 * r59415;
        double r59417 = r59411 - r59416;
        double r59418 = sqrt(r59417);
        double r59419 = r59410 - r59418;
        double r59420 = 2.0;
        double r59421 = r59420 * r59413;
        double r59422 = r59419 / r59421;
        return r59422;
}
↓
double f(double a, double b, double c) {
        double r59423 = b;
        double r59424 = -977083.9042033921;
        bool r59425 = r59423 <= r59424;
        double r59426 = -1.0;
        double r59427 = c;
        double r59428 = r59427 / r59423;
        double r59429 = r59426 * r59428;
        double r59430 = -2.2776246100069977e-291;
        bool r59431 = r59423 <= r59430;
        double r59432 = 1.0;
        double r59433 = 2.0;
        double r59434 = pow(r59423, r59433);
        double r59435 = r59434 - r59434;
        double r59436 = 4.0;
        double r59437 = a;
        double r59438 = r59437 * r59427;
        double r59439 = r59436 * r59438;
        double r59440 = r59435 + r59439;
        double r59441 = r59432 * r59440;
        double r59442 = 2.0;
        double r59443 = r59442 * r59437;
        double r59444 = r59441 / r59443;
        double r59445 = -r59423;
        double r59446 = r59423 * r59423;
        double r59447 = r59446 - r59439;
        double r59448 = sqrt(r59447);
        double r59449 = r59445 + r59448;
        double r59450 = r59444 / r59449;
        double r59451 = 4.2945314309789726e+104;
        bool r59452 = r59423 <= r59451;
        double r59453 = r59445 / r59443;
        double r59454 = r59448 / r59443;
        double r59455 = r59453 - r59454;
        double r59456 = r59423 / r59437;
        double r59457 = r59426 * r59456;
        double r59458 = r59452 ? r59455 : r59457;
        double r59459 = r59431 ? r59450 : r59458;
        double r59460 = r59425 ? r59429 : r59459;
        return r59460;
}

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

Original34.0
Target21.0
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -977083.9042033921

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 5.1

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

    if -977083.9042033921 < b < -2.2776246100069977e-291

    1. Initial program 27.2

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

    3. Applied clear-num27.3

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

    5. Applied flip--27.3

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

    6. Applied associate-/r/27.4

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

    7. Applied associate-/r*27.4

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

    8. Simplified17.2

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

    if -2.2776246100069977e-291 < b < 4.2945314309789726e+104

    1. Initial program 8.6

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

    3. Applied div-sub8.6

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

    if 4.2945314309789726e+104 < b

    1. Initial program 47.7

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

    3. Applied clear-num47.7

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

    4. Taylor expanded around 0 3.7

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -977083.9042033920995891094207763671875:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.277624610006997683151417691157213452138 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 4.294531430978972552500215519031197589784 \cdot 10^{104}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))