Average Error: 34.4 → 10.3
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \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 r101446 = b;
        double r101447 = -r101446;
        double r101448 = r101446 * r101446;
        double r101449 = 4.0;
        double r101450 = a;
        double r101451 = c;
        double r101452 = r101450 * r101451;
        double r101453 = r101449 * r101452;
        double r101454 = r101448 - r101453;
        double r101455 = sqrt(r101454);
        double r101456 = r101447 - r101455;
        double r101457 = 2.0;
        double r101458 = r101457 * r101450;
        double r101459 = r101456 / r101458;
        return r101459;
}
↓
double f(double a, double b, double c) {
        double r101460 = b;
        double r101461 = -1.0674124610604968e-82;
        bool r101462 = r101460 <= r101461;
        double r101463 = -1.0;
        double r101464 = c;
        double r101465 = r101464 / r101460;
        double r101466 = r101463 * r101465;
        double r101467 = 5.968766258400916e+107;
        bool r101468 = r101460 <= r101467;
        double r101469 = 1.0;
        double r101470 = -r101460;
        double r101471 = r101460 * r101460;
        double r101472 = 4.0;
        double r101473 = a;
        double r101474 = r101473 * r101464;
        double r101475 = r101472 * r101474;
        double r101476 = r101471 - r101475;
        double r101477 = sqrt(r101476);
        double r101478 = r101470 - r101477;
        double r101479 = 2.0;
        double r101480 = r101479 * r101473;
        double r101481 = r101478 / r101480;
        double r101482 = r101469 * r101481;
        double r101483 = 1.0;
        double r101484 = r101460 / r101473;
        double r101485 = r101465 - r101484;
        double r101486 = r101483 * r101485;
        double r101487 = r101468 ? r101482 : r101486;
        double r101488 = r101462 ? r101466 : r101487;
        return r101488;
}

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
Target21.5
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0674124610604968e-82

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.9

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

    if -1.0674124610604968e-82 < b < 5.968766258400916e+107

    1. Initial program 13.7

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

    3. Applied clear-num13.9

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

    5. Applied *-un-lft-identity13.9

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

    6. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Applied times-frac13.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    8. Simplified13.9

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified13.7

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

    if 5.968766258400916e+107 < b

    1. Initial program 50.0

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

    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 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))