Average Error: 32.8 → 10.3
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6239127264630285 \cdot 10^{-63}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.052614559736995 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \frac{\frac{-1}{2}}{a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.6239127264630285 \cdot 10^{-63}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 7.052614559736995 \cdot 10^{+62}:\\
\;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \frac{\frac{-1}{2}}{a} \cdot b\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3966556 = b;
        double r3966557 = -r3966556;
        double r3966558 = r3966556 * r3966556;
        double r3966559 = 4.0;
        double r3966560 = a;
        double r3966561 = c;
        double r3966562 = r3966560 * r3966561;
        double r3966563 = r3966559 * r3966562;
        double r3966564 = r3966558 - r3966563;
        double r3966565 = sqrt(r3966564);
        double r3966566 = r3966557 - r3966565;
        double r3966567 = 2.0;
        double r3966568 = r3966567 * r3966560;
        double r3966569 = r3966566 / r3966568;
        return r3966569;
}
↓
double f(double a, double b, double c) {
        double r3966570 = b;
        double r3966571 = -1.6239127264630285e-63;
        bool r3966572 = r3966570 <= r3966571;
        double r3966573 = c;
        double r3966574 = r3966573 / r3966570;
        double r3966575 = -r3966574;
        double r3966576 = 7.052614559736995e+62;
        bool r3966577 = r3966570 <= r3966576;
        double r3966578 = -0.5;
        double r3966579 = a;
        double r3966580 = r3966578 / r3966579;
        double r3966581 = r3966570 * r3966570;
        double r3966582 = 4.0;
        double r3966583 = r3966582 * r3966579;
        double r3966584 = r3966583 * r3966573;
        double r3966585 = r3966581 - r3966584;
        double r3966586 = sqrt(r3966585);
        double r3966587 = r3966580 * r3966586;
        double r3966588 = r3966580 * r3966570;
        double r3966589 = r3966587 + r3966588;
        double r3966590 = r3966570 / r3966579;
        double r3966591 = r3966574 - r3966590;
        double r3966592 = r3966577 ? r3966589 : r3966591;
        double r3966593 = r3966572 ? r3966575 : r3966592;
        return r3966593;
}

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

Original32.8
Target20.1
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6239127264630285e-63

    1. Initial program 52.2

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

    2. Taylor expanded around -inf 8.6

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

    3. Simplified8.6

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

    if -1.6239127264630285e-63 < b < 7.052614559736995e+62

    1. Initial program 13.9

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

    3. Applied clear-num14.0

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

    5. Applied div-inv14.1

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

    6. Applied add-cube-cbrt14.1

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

    7. Applied times-frac14.1

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

    8. Simplified14.1

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

    9. Simplified14.0

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

    11. Applied distribute-neg-in14.0

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

    12. Applied distribute-lft-in14.1

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

    if 7.052614559736995e+62 < b

    1. Initial program 38.1

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

    2. Taylor expanded around inf 4.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6239127264630285 \cdot 10^{-63}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.052614559736995 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \frac{\frac{-1}{2}}{a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))