Average Error: 34.5 → 8.7
Time: 20.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.997712787466911997482638512886606028789 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -9.997712787466911997482638512886606028789 \cdot 10^{-253}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \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 r75081 = b;
        double r75082 = -r75081;
        double r75083 = r75081 * r75081;
        double r75084 = 4.0;
        double r75085 = a;
        double r75086 = c;
        double r75087 = r75085 * r75086;
        double r75088 = r75084 * r75087;
        double r75089 = r75083 - r75088;
        double r75090 = sqrt(r75089);
        double r75091 = r75082 - r75090;
        double r75092 = 2.0;
        double r75093 = r75092 * r75085;
        double r75094 = r75091 / r75093;
        return r75094;
}
↓
double f(double a, double b, double c) {
        double r75095 = b;
        double r75096 = -3.1028950157805323e+69;
        bool r75097 = r75095 <= r75096;
        double r75098 = -1.0;
        double r75099 = c;
        double r75100 = r75099 / r75095;
        double r75101 = r75098 * r75100;
        double r75102 = -9.997712787466912e-253;
        bool r75103 = r75095 <= r75102;
        double r75104 = 4.0;
        double r75105 = a;
        double r75106 = r75104 * r75105;
        double r75107 = r75106 * r75099;
        double r75108 = 2.0;
        double r75109 = r75108 * r75105;
        double r75110 = r75107 / r75109;
        double r75111 = 2.0;
        double r75112 = pow(r75095, r75111);
        double r75113 = r75105 * r75099;
        double r75114 = r75104 * r75113;
        double r75115 = r75112 - r75114;
        double r75116 = sqrt(r75115);
        double r75117 = r75116 - r75095;
        double r75118 = r75110 / r75117;
        double r75119 = 2.1255630798514387e+135;
        bool r75120 = r75095 <= r75119;
        double r75121 = -r75095;
        double r75122 = r75121 / r75109;
        double r75123 = r75095 * r75095;
        double r75124 = r75123 - r75114;
        double r75125 = sqrt(r75124);
        double r75126 = r75125 / r75109;
        double r75127 = r75122 - r75126;
        double r75128 = 1.0;
        double r75129 = r75095 / r75105;
        double r75130 = r75100 - r75129;
        double r75131 = r75128 * r75130;
        double r75132 = r75120 ? r75127 : r75131;
        double r75133 = r75103 ? r75118 : r75132;
        double r75134 = r75097 ? r75101 : r75133;
        return r75134;
}

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

Original34.5
Target21.3
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1028950157805323e+69

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.1

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

    if -3.1028950157805323e+69 < b < -9.997712787466912e-253

    1. Initial program 33.1

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

    3. Applied flip--33.1

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    7. Applied div-inv16.9

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

    9. Applied pow116.9

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

    10. Applied pow116.9

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

    11. Applied pow-prod-down16.9

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

    12. Simplified16.3

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

    if -9.997712787466912e-253 < b < 2.1255630798514387e+135

    1. Initial program 9.6

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

    3. Applied div-sub9.6

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

    if 2.1255630798514387e+135 < b

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.997712787466911997482638512886606028789 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))