Average Error: 34.1 → 10.5
Time: 22.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\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 r2850586 = b;
        double r2850587 = -r2850586;
        double r2850588 = r2850586 * r2850586;
        double r2850589 = 4.0;
        double r2850590 = a;
        double r2850591 = c;
        double r2850592 = r2850590 * r2850591;
        double r2850593 = r2850589 * r2850592;
        double r2850594 = r2850588 - r2850593;
        double r2850595 = sqrt(r2850594);
        double r2850596 = r2850587 - r2850595;
        double r2850597 = 2.0;
        double r2850598 = r2850597 * r2850590;
        double r2850599 = r2850596 / r2850598;
        return r2850599;
}
↓
double f(double a, double b, double c) {
        double r2850600 = b;
        double r2850601 = -9.332433396832084e-58;
        bool r2850602 = r2850600 <= r2850601;
        double r2850603 = -1.0;
        double r2850604 = c;
        double r2850605 = r2850604 / r2850600;
        double r2850606 = r2850603 * r2850605;
        double r2850607 = 3.038903409991338e+107;
        bool r2850608 = r2850600 <= r2850607;
        double r2850609 = 2.0;
        double r2850610 = a;
        double r2850611 = r2850609 * r2850610;
        double r2850612 = r2850600 / r2850611;
        double r2850613 = -r2850612;
        double r2850614 = r2850600 * r2850600;
        double r2850615 = 4.0;
        double r2850616 = r2850610 * r2850604;
        double r2850617 = r2850615 * r2850616;
        double r2850618 = r2850614 - r2850617;
        double r2850619 = sqrt(r2850618);
        double r2850620 = r2850619 / r2850611;
        double r2850621 = r2850613 - r2850620;
        double r2850622 = 1.0;
        double r2850623 = r2850600 / r2850610;
        double r2850624 = r2850605 - r2850623;
        double r2850625 = r2850622 * r2850624;
        double r2850626 = r2850608 ? r2850621 : r2850625;
        double r2850627 = r2850602 ? r2850606 : r2850626;
        return r2850627;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b < 3.038903409991338e+107

    1. Initial program 14.1

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

    3. Applied div-sub14.1

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

    if 3.038903409991338e+107 < b

    1. Initial program 49.2

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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)))