Average Error: 33.6 → 7.8
Time: 26.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.8260933955440565 \cdot 10^{-16}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -1.0403213044374248 \cdot 10^{-202}:\\ \;\;\;\;\frac{c}{\frac{2 \cdot a}{\frac{a \cdot 4}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}}\\ \mathbf{elif}\;b \le 4.738941069295542 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.8260933955440565 \cdot 10^{-16}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -1.0403213044374248 \cdot 10^{-202}:\\
\;\;\;\;\frac{c}{\frac{2 \cdot a}{\frac{a \cdot 4}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}}\\

\mathbf{elif}\;b \le 4.738941069295542 \cdot 10^{+124}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3706063 = b;
        double r3706064 = -r3706063;
        double r3706065 = r3706063 * r3706063;
        double r3706066 = 4.0;
        double r3706067 = a;
        double r3706068 = c;
        double r3706069 = r3706067 * r3706068;
        double r3706070 = r3706066 * r3706069;
        double r3706071 = r3706065 - r3706070;
        double r3706072 = sqrt(r3706071);
        double r3706073 = r3706064 - r3706072;
        double r3706074 = 2.0;
        double r3706075 = r3706074 * r3706067;
        double r3706076 = r3706073 / r3706075;
        return r3706076;
}
↓
double f(double a, double b, double c) {
        double r3706077 = b;
        double r3706078 = -3.8260933955440565e-16;
        bool r3706079 = r3706077 <= r3706078;
        double r3706080 = c;
        double r3706081 = r3706080 / r3706077;
        double r3706082 = -r3706081;
        double r3706083 = -1.0403213044374248e-202;
        bool r3706084 = r3706077 <= r3706083;
        double r3706085 = 2.0;
        double r3706086 = a;
        double r3706087 = r3706085 * r3706086;
        double r3706088 = 4.0;
        double r3706089 = r3706086 * r3706088;
        double r3706090 = r3706077 * r3706077;
        double r3706091 = r3706089 * r3706080;
        double r3706092 = r3706090 - r3706091;
        double r3706093 = sqrt(r3706092);
        double r3706094 = r3706093 - r3706077;
        double r3706095 = r3706089 / r3706094;
        double r3706096 = r3706087 / r3706095;
        double r3706097 = r3706080 / r3706096;
        double r3706098 = 4.738941069295542e+124;
        bool r3706099 = r3706077 <= r3706098;
        double r3706100 = -r3706077;
        double r3706101 = r3706080 * r3706086;
        double r3706102 = r3706088 * r3706101;
        double r3706103 = r3706090 - r3706102;
        double r3706104 = sqrt(r3706103);
        double r3706105 = r3706100 - r3706104;
        double r3706106 = r3706105 / r3706087;
        double r3706107 = r3706077 / r3706086;
        double r3706108 = r3706081 - r3706107;
        double r3706109 = r3706099 ? r3706106 : r3706108;
        double r3706110 = r3706084 ? r3706097 : r3706109;
        double r3706111 = r3706079 ? r3706082 : r3706110;
        return r3706111;
}

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

Original33.6
Target20.8
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.8260933955440565e-16

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 6.5

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

    3. Simplified6.5

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

    if -3.8260933955440565e-16 < b < -1.0403213044374248e-202

    1. Initial program 28.6

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

    3. Applied add-sqr-sqrt28.6

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

    4. Applied sqrt-prod28.9

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

    6. Applied flip--29.0

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

    7. Simplified17.7

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

    8. Simplified17.5

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

    10. Applied *-un-lft-identity17.5

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

    11. Applied times-frac13.0

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

    12. Applied associate-/l*7.9

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

    if -1.0403213044374248e-202 < b < 4.738941069295542e+124

    1. Initial program 10.6

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

    if 4.738941069295542e+124 < b

    1. Initial program 50.6

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.8260933955440565 \cdot 10^{-16}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -1.0403213044374248 \cdot 10^{-202}:\\ \;\;\;\;\frac{c}{\frac{2 \cdot a}{\frac{a \cdot 4}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}}\\ \mathbf{elif}\;b \le 4.738941069295542 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))