Average Error: 34.5 → 10.2
Time: 4.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -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(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r84195 = b;
        double r84196 = -r84195;
        double r84197 = r84195 * r84195;
        double r84198 = 4.0;
        double r84199 = a;
        double r84200 = c;
        double r84201 = r84199 * r84200;
        double r84202 = r84198 * r84201;
        double r84203 = r84197 - r84202;
        double r84204 = sqrt(r84203);
        double r84205 = r84196 - r84204;
        double r84206 = 2.0;
        double r84207 = r84206 * r84199;
        double r84208 = r84205 / r84207;
        return r84208;
}
↓
double f(double a, double b, double c) {
        double r84209 = b;
        double r84210 = -4.706781135059312e-92;
        bool r84211 = r84209 <= r84210;
        double r84212 = 1.0;
        double r84213 = -1.0;
        double r84214 = c;
        double r84215 = r84214 / r84209;
        double r84216 = r84213 * r84215;
        double r84217 = r84212 * r84216;
        double r84218 = 5.722235152988638e+98;
        bool r84219 = r84209 <= r84218;
        double r84220 = -r84209;
        double r84221 = r84209 * r84209;
        double r84222 = 4.0;
        double r84223 = a;
        double r84224 = r84223 * r84214;
        double r84225 = r84222 * r84224;
        double r84226 = r84221 - r84225;
        double r84227 = sqrt(r84226);
        double r84228 = r84220 - r84227;
        double r84229 = 2.0;
        double r84230 = r84229 * r84223;
        double r84231 = r84212 / r84230;
        double r84232 = r84228 * r84231;
        double r84233 = 1.0;
        double r84234 = r84209 / r84223;
        double r84235 = r84215 - r84234;
        double r84236 = r84233 * r84235;
        double r84237 = r84219 ? r84232 : r84236;
        double r84238 = r84211 ? r84217 : r84237;
        return r84238;
}

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. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification10.2

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

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)))