Average Error: 34.2 → 6.8
Time: 4.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.40284932349203652 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\ \mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.40284932349203652 \cdot 10^{128}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\
\;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\

\mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\
\;\;\;\;\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 r98124 = b;
        double r98125 = -r98124;
        double r98126 = r98124 * r98124;
        double r98127 = 4.0;
        double r98128 = a;
        double r98129 = c;
        double r98130 = r98128 * r98129;
        double r98131 = r98127 * r98130;
        double r98132 = r98126 - r98131;
        double r98133 = sqrt(r98132);
        double r98134 = r98125 - r98133;
        double r98135 = 2.0;
        double r98136 = r98135 * r98128;
        double r98137 = r98134 / r98136;
        return r98137;
}
↓
double f(double a, double b, double c) {
        double r98138 = b;
        double r98139 = -2.4028493234920365e+128;
        bool r98140 = r98138 <= r98139;
        double r98141 = -1.0;
        double r98142 = c;
        double r98143 = r98142 / r98138;
        double r98144 = r98141 * r98143;
        double r98145 = 5.877669040907696e-167;
        bool r98146 = r98138 <= r98145;
        double r98147 = 2.0;
        double r98148 = r98147 * r98142;
        double r98149 = r98138 * r98138;
        double r98150 = 4.0;
        double r98151 = a;
        double r98152 = r98151 * r98142;
        double r98153 = r98150 * r98152;
        double r98154 = r98149 - r98153;
        double r98155 = sqrt(r98154);
        double r98156 = r98155 - r98138;
        double r98157 = r98148 / r98156;
        double r98158 = 1.0;
        double r98159 = pow(r98157, r98158);
        double r98160 = 1.5849721394456554e+84;
        bool r98161 = r98138 <= r98160;
        double r98162 = -r98138;
        double r98163 = r98162 - r98155;
        double r98164 = r98147 * r98151;
        double r98165 = r98163 / r98164;
        double r98166 = 1.0;
        double r98167 = r98138 / r98151;
        double r98168 = r98143 - r98167;
        double r98169 = r98166 * r98168;
        double r98170 = r98161 ? r98165 : r98169;
        double r98171 = r98146 ? r98159 : r98170;
        double r98172 = r98140 ? r98144 : r98171;
        return r98172;
}

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

Original34.2
Target21.2
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.4028493234920365e+128

    1. Initial program 61.5

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

    2. Taylor expanded around -inf 2.2

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

    if -2.4028493234920365e+128 < b < 5.877669040907696e-167

    1. Initial program 29.5

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

    3. Applied div-inv29.6

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

    5. Applied flip--29.8

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

    6. Simplified16.1

      \[\leadsto \frac{\color{blue}{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{2 \cdot a}\]

    7. Simplified16.1

      \[\leadsto \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{2 \cdot a}\]
    8. Using strategy rm

    9. Applied pow116.1

      \[\leadsto \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{1}}\]

    10. Applied pow116.1

      \[\leadsto \color{blue}{{\left(\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}} \cdot {\left(\frac{1}{2 \cdot a}\right)}^{1}\]

    11. Applied pow-prod-down16.1

      \[\leadsto \color{blue}{{\left(\frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\right)}^{1}}\]

    12. Simplified15.0

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

    13. Taylor expanded around 0 10.0

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

    if 5.877669040907696e-167 < b < 1.5849721394456554e+84

    1. Initial program 7.0

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

    if 1.5849721394456554e+84 < b

    1. Initial program 43.1

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.40284932349203652 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\ \mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))