Average Error: 34.1 → 9.4
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\
\;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r75496 = b;
        double r75497 = -r75496;
        double r75498 = r75496 * r75496;
        double r75499 = 4.0;
        double r75500 = a;
        double r75501 = c;
        double r75502 = r75500 * r75501;
        double r75503 = r75499 * r75502;
        double r75504 = r75498 - r75503;
        double r75505 = sqrt(r75504);
        double r75506 = r75497 - r75505;
        double r75507 = 2.0;
        double r75508 = r75507 * r75500;
        double r75509 = r75506 / r75508;
        return r75509;
}
↓
double f(double a, double b, double c) {
        double r75510 = b;
        double r75511 = -9.913936217078672e+60;
        bool r75512 = r75510 <= r75511;
        double r75513 = -1.0;
        double r75514 = c;
        double r75515 = r75514 / r75510;
        double r75516 = r75513 * r75515;
        double r75517 = -2.8768523970510503e-50;
        bool r75518 = r75510 <= r75517;
        double r75519 = 4.0;
        double r75520 = a;
        double r75521 = r75519 * r75520;
        double r75522 = r75514 * r75521;
        double r75523 = r75510 * r75510;
        double r75524 = r75521 * r75514;
        double r75525 = -r75524;
        double r75526 = r75523 + r75525;
        double r75527 = sqrt(r75526);
        double r75528 = r75527 - r75510;
        double r75529 = r75522 / r75528;
        double r75530 = 2.0;
        double r75531 = r75530 * r75520;
        double r75532 = r75529 / r75531;
        double r75533 = -1.1244234173654837e-68;
        bool r75534 = r75510 <= r75533;
        double r75535 = 9.28238754477782e+92;
        bool r75536 = r75510 <= r75535;
        double r75537 = -r75510;
        double r75538 = r75520 * r75514;
        double r75539 = r75519 * r75538;
        double r75540 = r75523 - r75539;
        double r75541 = sqrt(r75540);
        double r75542 = r75537 - r75541;
        double r75543 = r75542 / r75530;
        double r75544 = r75543 / r75520;
        double r75545 = -2.0;
        double r75546 = r75545 * r75510;
        double r75547 = r75546 / r75531;
        double r75548 = r75536 ? r75544 : r75547;
        double r75549 = r75534 ? r75516 : r75548;
        double r75550 = r75518 ? r75532 : r75549;
        double r75551 = r75512 ? r75516 : r75550;
        return r75551;
}

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

Original34.1
Target20.8
Herbie9.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.913936217078672e+60 or -2.8768523970510503e-50 < b < -1.1244234173654837e-68

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 5.0

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

    if -9.913936217078672e+60 < b < -2.8768523970510503e-50

    1. Initial program 43.8

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

    3. Applied flip--43.9

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

    4. Simplified13.5

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

    5. Simplified13.5

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

    7. Applied sub-neg13.5

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

    8. Simplified13.5

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

    if -1.1244234173654837e-68 < b < 9.28238754477782e+92

    1. Initial program 13.4

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

    3. Applied associate-/r*13.4

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

    if 9.28238754477782e+92 < b

    1. Initial program 45.0

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

    3. Applied flip--62.8

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

    4. Simplified61.9

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

    5. Simplified61.9

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

    6. Taylor expanded around 0 3.6

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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)))