Average Error: 33.1 → 9.2
Time: 1.1m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\
\;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\
\;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r32132041 = b;
        double r32132042 = -r32132041;
        double r32132043 = r32132041 * r32132041;
        double r32132044 = 4.0;
        double r32132045 = a;
        double r32132046 = r32132044 * r32132045;
        double r32132047 = c;
        double r32132048 = r32132046 * r32132047;
        double r32132049 = r32132043 - r32132048;
        double r32132050 = sqrt(r32132049);
        double r32132051 = r32132042 + r32132050;
        double r32132052 = 2.0;
        double r32132053 = r32132052 * r32132045;
        double r32132054 = r32132051 / r32132053;
        return r32132054;
}
↓
double f(double a, double b, double c) {
        double r32132055 = b;
        double r32132056 = -1.6124744946043857e+143;
        bool r32132057 = r32132055 <= r32132056;
        double r32132058 = c;
        double r32132059 = r32132058 / r32132055;
        double r32132060 = a;
        double r32132061 = r32132055 / r32132060;
        double r32132062 = r32132059 - r32132061;
        double r32132063 = 1.754487595174753e-113;
        bool r32132064 = r32132055 <= r32132063;
        double r32132065 = -4.0;
        double r32132066 = r32132058 * r32132065;
        double r32132067 = r32132066 * r32132060;
        double r32132068 = r32132055 * r32132055;
        double r32132069 = r32132067 + r32132068;
        double r32132070 = sqrt(r32132069);
        double r32132071 = r32132070 - r32132055;
        double r32132072 = 2.0;
        double r32132073 = r32132060 * r32132072;
        double r32132074 = r32132071 / r32132073;
        double r32132075 = 9.873738165909194e+16;
        bool r32132076 = r32132055 <= r32132075;
        double r32132077 = 0.5;
        double r32132078 = r32132077 / r32132060;
        double r32132079 = r32132066 * r32132078;
        double r32132080 = r32132060 * r32132079;
        double r32132081 = r32132070 + r32132055;
        double r32132082 = r32132080 / r32132081;
        double r32132083 = -r32132059;
        double r32132084 = r32132076 ? r32132082 : r32132083;
        double r32132085 = r32132064 ? r32132074 : r32132084;
        double r32132086 = r32132057 ? r32132062 : r32132085;
        return r32132086;
}

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

Original33.1
Target20.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6124744946043857e+143

    1. Initial program 57.1

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

    2. Simplified57.1

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

    3. Taylor expanded around -inf 2.6

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

    if -1.6124744946043857e+143 < b < 1.754487595174753e-113

    1. Initial program 11.1

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

    2. Simplified11.1

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

    3. Taylor expanded around -inf 11.1

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

    4. Simplified11.1

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

    if 1.754487595174753e-113 < b < 9.873738165909194e+16

    1. Initial program 38.5

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

    2. Simplified38.5

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

    3. Taylor expanded around -inf 38.5

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

    4. Simplified38.5

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

    6. Applied clear-num38.5

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

    8. Applied flip--38.6

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

    9. Applied associate-/r/38.6

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

    10. Applied associate-/r*38.6

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

    11. Simplified18.7

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

    if 9.873738165909194e+16 < b

    1. Initial program 54.7

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

    2. Simplified54.7

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

    3. Taylor expanded around -inf 54.7

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

    4. Simplified54.7

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

    5. Taylor expanded around inf 5.7

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

    6. Simplified5.7

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))