Average Error: 34.5 → 10.2
Time: 4.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]
\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 -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\
\;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85492 = b;
        double r85493 = -r85492;
        double r85494 = r85492 * r85492;
        double r85495 = 4.0;
        double r85496 = a;
        double r85497 = c;
        double r85498 = r85496 * r85497;
        double r85499 = r85495 * r85498;
        double r85500 = r85494 - r85499;
        double r85501 = sqrt(r85500);
        double r85502 = r85493 - r85501;
        double r85503 = 2.0;
        double r85504 = r85503 * r85496;
        double r85505 = r85502 / r85504;
        return r85505;
}
↓
double f(double a, double b, double c) {
        double r85506 = b;
        double r85507 = -4.706781135059312e-92;
        bool r85508 = r85506 <= r85507;
        double r85509 = 1.0;
        double r85510 = -1.0;
        double r85511 = c;
        double r85512 = r85511 / r85506;
        double r85513 = r85510 * r85512;
        double r85514 = r85509 * r85513;
        double r85515 = 5.722235152988638e+98;
        bool r85516 = r85506 <= r85515;
        double r85517 = -r85506;
        double r85518 = r85506 * r85506;
        double r85519 = 4.0;
        double r85520 = a;
        double r85521 = r85520 * r85511;
        double r85522 = r85519 * r85521;
        double r85523 = r85518 - r85522;
        double r85524 = sqrt(r85523);
        double r85525 = r85517 - r85524;
        double r85526 = 2.0;
        double r85527 = r85526 * r85520;
        double r85528 = r85509 / r85527;
        double r85529 = r85525 * r85528;
        double r85530 = 1.0;
        double r85531 = r85506 / r85520;
        double r85532 = r85512 - r85531;
        double r85533 = r85530 * r85532;
        double r85534 = r85509 * r85533;
        double r85535 = r85516 ? r85529 : r85534;
        double r85536 = r85508 ? r85514 : r85535;
        return r85536;
}

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

Original34.5
Target21.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.706781135059312e-92

    1. Initial program 52.4

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

    3. Applied div-inv52.4

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

    5. Applied *-un-lft-identity52.4

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

    6. Applied associate-*l*52.4

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

    7. Simplified52.4

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

    8. Taylor expanded around -inf 10.3

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

    if -4.706781135059312e-92 < b < 5.722235152988638e+98

    1. Initial program 12.7

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

    3. Applied div-inv12.8

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

    if 5.722235152988638e+98 < b

    1. Initial program 47.2

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

    3. Applied div-inv47.2

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

    5. Applied *-un-lft-identity47.2

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

    6. Applied associate-*l*47.2

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

    7. Simplified47.2

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

    8. Taylor expanded around inf 3.6

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

    9. Simplified3.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))