Average Error: 34.4 → 8.4
Time: 43.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1927067580012775 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\frac{\left(a \cdot c\right) \cdot 4.0}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}}{2.0 \cdot a}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;-\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + b}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;-1.0 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -6.1927067580012775 \cdot 10^{-115}:\\
\;\;\;\;\frac{-\frac{\left(a \cdot c\right) \cdot 4.0}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}}{2.0 \cdot a}\\

\mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\
\;\;\;\;-\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + b}{2.0 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\

\end{array}
double f(double a, double b, double c) {
        double r4810417 = b;
        double r4810418 = -r4810417;
        double r4810419 = r4810417 * r4810417;
        double r4810420 = 4.0;
        double r4810421 = a;
        double r4810422 = c;
        double r4810423 = r4810421 * r4810422;
        double r4810424 = r4810420 * r4810423;
        double r4810425 = r4810419 - r4810424;
        double r4810426 = sqrt(r4810425);
        double r4810427 = r4810418 - r4810426;
        double r4810428 = 2.0;
        double r4810429 = r4810428 * r4810421;
        double r4810430 = r4810427 / r4810429;
        return r4810430;
}
↓
double f(double a, double b, double c) {
        double r4810431 = b;
        double r4810432 = -3.234164035284793e+22;
        bool r4810433 = r4810431 <= r4810432;
        double r4810434 = -1.0;
        double r4810435 = c;
        double r4810436 = r4810435 / r4810431;
        double r4810437 = r4810434 * r4810436;
        double r4810438 = -6.1927067580012775e-115;
        bool r4810439 = r4810431 <= r4810438;
        double r4810440 = a;
        double r4810441 = r4810440 * r4810435;
        double r4810442 = 4.0;
        double r4810443 = r4810441 * r4810442;
        double r4810444 = r4810431 * r4810431;
        double r4810445 = r4810444 - r4810443;
        double r4810446 = sqrt(r4810445);
        double r4810447 = r4810431 - r4810446;
        double r4810448 = r4810443 / r4810447;
        double r4810449 = -r4810448;
        double r4810450 = 2.0;
        double r4810451 = r4810450 * r4810440;
        double r4810452 = r4810449 / r4810451;
        double r4810453 = 2.026128983134594e+103;
        bool r4810454 = r4810431 <= r4810453;
        double r4810455 = r4810446 + r4810431;
        double r4810456 = r4810455 / r4810451;
        double r4810457 = -r4810456;
        double r4810458 = r4810431 / r4810440;
        double r4810459 = r4810436 - r4810458;
        double r4810460 = 1.0;
        double r4810461 = r4810459 * r4810460;
        double r4810462 = r4810454 ? r4810457 : r4810461;
        double r4810463 = r4810439 ? r4810452 : r4810462;
        double r4810464 = r4810433 ? r4810437 : r4810463;
        return r4810464;
}

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

Original34.4
Target21.3
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.234164035284793e+22

    1. Initial program 56.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]

    2. Taylor expanded around -inf 4.6

      \[\leadsto \color{blue}{-1.0 \cdot \frac{c}{b}}\]

    if -3.234164035284793e+22 < b < -6.1927067580012775e-115

    1. Initial program 38.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv38.6

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2.0 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/38.6

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot 1}{2.0 \cdot a}}\]

    6. Simplified38.6

      \[\leadsto \frac{\color{blue}{-\left(b + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right)}}{2.0 \cdot a}\]
    7. Using strategy rm

    8. Applied flip-+38.6

      \[\leadsto \frac{-\color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}}}{2.0 \cdot a}\]

    9. Simplified15.8

      \[\leadsto \frac{-\frac{\color{blue}{0 + 4.0 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}}{2.0 \cdot a}\]

    if -6.1927067580012775e-115 < b < 2.026128983134594e+103

    1. Initial program 11.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.6

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2.0 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/11.5

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot 1}{2.0 \cdot a}}\]

    6. Simplified11.5

      \[\leadsto \frac{\color{blue}{-\left(b + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right)}}{2.0 \cdot a}\]

    if 2.026128983134594e+103 < b

    1. Initial program 47.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]

    2. Taylor expanded around inf 3.2

      \[\leadsto \color{blue}{1.0 \cdot \frac{c}{b} - 1.0 \cdot \frac{b}{a}}\]

    3. Simplified3.2

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1927067580012775 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\frac{\left(a \cdot c\right) \cdot 4.0}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}}{2.0 \cdot a}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;-\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + b}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))