Average Error: 33.9 → 6.5
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.426750935011225297811866513376863104463 \cdot 10^{149}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.085606243292902438128577538806679181907 \cdot 10^{-172}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.631578693473388191161659038702797063909 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.426750935011225297811866513376863104463 \cdot 10^{149}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.085606243292902438128577538806679181907 \cdot 10^{-172}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 3.631578693473388191161659038702797063909 \cdot 10^{102}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r84496 = b;
        double r84497 = -r84496;
        double r84498 = r84496 * r84496;
        double r84499 = 4.0;
        double r84500 = a;
        double r84501 = c;
        double r84502 = r84500 * r84501;
        double r84503 = r84499 * r84502;
        double r84504 = r84498 - r84503;
        double r84505 = sqrt(r84504);
        double r84506 = r84497 - r84505;
        double r84507 = 2.0;
        double r84508 = r84507 * r84500;
        double r84509 = r84506 / r84508;
        return r84509;
}
↓
double f(double a, double b, double c) {
        double r84510 = b;
        double r84511 = -1.4267509350112253e+149;
        bool r84512 = r84510 <= r84511;
        double r84513 = -1.0;
        double r84514 = c;
        double r84515 = r84514 / r84510;
        double r84516 = r84513 * r84515;
        double r84517 = 1.0856062432929024e-172;
        bool r84518 = r84510 <= r84517;
        double r84519 = 2.0;
        double r84520 = r84519 * r84514;
        double r84521 = r84510 * r84510;
        double r84522 = 4.0;
        double r84523 = a;
        double r84524 = r84523 * r84514;
        double r84525 = r84522 * r84524;
        double r84526 = r84521 - r84525;
        double r84527 = sqrt(r84526);
        double r84528 = r84527 - r84510;
        double r84529 = r84520 / r84528;
        double r84530 = 3.631578693473388e+102;
        bool r84531 = r84510 <= r84530;
        double r84532 = -r84510;
        double r84533 = sqrt(r84527);
        double r84534 = r84533 * r84533;
        double r84535 = r84532 - r84534;
        double r84536 = r84519 * r84523;
        double r84537 = r84535 / r84536;
        double r84538 = 1.0;
        double r84539 = r84510 / r84523;
        double r84540 = r84515 - r84539;
        double r84541 = r84538 * r84540;
        double r84542 = r84531 ? r84537 : r84541;
        double r84543 = r84518 ? r84529 : r84542;
        double r84544 = r84512 ? r84516 : r84543;
        return r84544;
}

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

Original33.9
Target20.9
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.4267509350112253e+149

    1. Initial program 63.3

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

    2. Taylor expanded around -inf 1.4

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

    if -1.4267509350112253e+149 < b < 1.0856062432929024e-172

    1. Initial program 30.8

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

    3. Applied div-inv30.8

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

    5. Applied flip--31.0

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

    6. Simplified15.6

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

    7. Simplified15.6

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

    9. Applied associate-*l/14.4

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

    10. Simplified14.3

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

    11. Taylor expanded around 0 9.5

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

    if 1.0856062432929024e-172 < b < 3.631578693473388e+102

    1. Initial program 6.4

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

    3. Applied add-sqr-sqrt6.4

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

    4. Applied sqrt-prod6.6

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

    if 3.631578693473388e+102 < b

    1. Initial program 47.0

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.426750935011225297811866513376863104463 \cdot 10^{149}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.085606243292902438128577538806679181907 \cdot 10^{-172}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.631578693473388191161659038702797063909 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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)))