Average Error: 33.2 → 9.8
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1629070 = b;
        double r1629071 = -r1629070;
        double r1629072 = r1629070 * r1629070;
        double r1629073 = 4.0;
        double r1629074 = a;
        double r1629075 = c;
        double r1629076 = r1629074 * r1629075;
        double r1629077 = r1629073 * r1629076;
        double r1629078 = r1629072 - r1629077;
        double r1629079 = sqrt(r1629078);
        double r1629080 = r1629071 - r1629079;
        double r1629081 = 2.0;
        double r1629082 = r1629081 * r1629074;
        double r1629083 = r1629080 / r1629082;
        return r1629083;
}
↓
double f(double a, double b, double c) {
        double r1629084 = b;
        double r1629085 = -1.8774910265390396e-73;
        bool r1629086 = r1629084 <= r1629085;
        double r1629087 = c;
        double r1629088 = r1629087 / r1629084;
        double r1629089 = -r1629088;
        double r1629090 = 2.5703497435733685e+102;
        bool r1629091 = r1629084 <= r1629090;
        double r1629092 = r1629084 * r1629084;
        double r1629093 = a;
        double r1629094 = r1629087 * r1629093;
        double r1629095 = 4.0;
        double r1629096 = r1629094 * r1629095;
        double r1629097 = r1629092 - r1629096;
        double r1629098 = sqrt(r1629097);
        double r1629099 = r1629098 + r1629084;
        double r1629100 = -r1629099;
        double r1629101 = 2.0;
        double r1629102 = r1629093 * r1629101;
        double r1629103 = r1629100 / r1629102;
        double r1629104 = r1629084 / r1629093;
        double r1629105 = r1629088 - r1629104;
        double r1629106 = r1629091 ? r1629103 : r1629105;
        double r1629107 = r1629086 ? r1629089 : r1629106;
        return r1629107;
}

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

Original33.2
Target20.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8774910265390396e-73

    1. Initial program 52.5

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

    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 -1.8774910265390396e-73 < b < 2.5703497435733685e+102

    1. Initial program 13.1

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

    3. Applied add-sqr-sqrt13.1

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

    4. Applied sqrt-prod13.3

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

    6. Applied neg-sub013.3

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

    7. Applied associate--l-13.3

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

    8. Simplified13.1

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

    if 2.5703497435733685e+102 < b

    1. Initial program 43.9

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))