Average Error: 34.0 → 10.6
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79557 = b;
        double r79558 = -r79557;
        double r79559 = r79557 * r79557;
        double r79560 = 4.0;
        double r79561 = a;
        double r79562 = c;
        double r79563 = r79561 * r79562;
        double r79564 = r79560 * r79563;
        double r79565 = r79559 - r79564;
        double r79566 = sqrt(r79565);
        double r79567 = r79558 - r79566;
        double r79568 = 2.0;
        double r79569 = r79568 * r79561;
        double r79570 = r79567 / r79569;
        return r79570;
}
↓
double f(double a, double b, double c) {
        double r79571 = b;
        double r79572 = -2.731633690849518e-121;
        bool r79573 = r79571 <= r79572;
        double r79574 = -1.0;
        double r79575 = c;
        double r79576 = r79575 / r79571;
        double r79577 = r79574 * r79576;
        double r79578 = 1.0273828621120979e+63;
        bool r79579 = r79571 <= r79578;
        double r79580 = 1.0;
        double r79581 = 2.0;
        double r79582 = a;
        double r79583 = r79581 * r79582;
        double r79584 = -r79571;
        double r79585 = r79571 * r79571;
        double r79586 = 4.0;
        double r79587 = r79582 * r79575;
        double r79588 = r79586 * r79587;
        double r79589 = r79585 - r79588;
        double r79590 = sqrt(r79589);
        double r79591 = r79584 - r79590;
        double r79592 = r79583 / r79591;
        double r79593 = r79580 / r79592;
        double r79594 = 1.0;
        double r79595 = r79571 / r79582;
        double r79596 = r79576 - r79595;
        double r79597 = r79594 * r79596;
        double r79598 = r79579 ? r79593 : r79597;
        double r79599 = r79573 ? r79577 : r79598;
        return r79599;
}

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

Original34.0
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.731633690849518e-121

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 11.5

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

    if -2.731633690849518e-121 < b < 1.0273828621120979e+63

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    if 1.0273828621120979e+63 < b

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))