Average Error: 33.8 → 9.3
Time: 18.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -0.03099989563658142946445117615894560003653:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\
\;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\

\mathbf{elif}\;b \le 63580190853209333432320:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r4757082 = b;
        double r4757083 = -r4757082;
        double r4757084 = r4757082 * r4757082;
        double r4757085 = 4.0;
        double r4757086 = a;
        double r4757087 = c;
        double r4757088 = r4757086 * r4757087;
        double r4757089 = r4757085 * r4757088;
        double r4757090 = r4757084 - r4757089;
        double r4757091 = sqrt(r4757090);
        double r4757092 = r4757083 - r4757091;
        double r4757093 = 2.0;
        double r4757094 = r4757093 * r4757086;
        double r4757095 = r4757092 / r4757094;
        return r4757095;
}
↓
double f(double a, double b, double c) {
        double r4757096 = b;
        double r4757097 = -0.03099989563658143;
        bool r4757098 = r4757096 <= r4757097;
        double r4757099 = -1.0;
        double r4757100 = c;
        double r4757101 = r4757100 / r4757096;
        double r4757102 = r4757099 * r4757101;
        double r4757103 = 5.992812285264993e-289;
        bool r4757104 = r4757096 <= r4757103;
        double r4757105 = 4.0;
        double r4757106 = r4757100 * r4757105;
        double r4757107 = a;
        double r4757108 = r4757106 * r4757107;
        double r4757109 = r4757096 * r4757096;
        double r4757110 = r4757109 - r4757108;
        double r4757111 = sqrt(r4757110);
        double r4757112 = r4757111 - r4757096;
        double r4757113 = r4757108 / r4757112;
        double r4757114 = 2.0;
        double r4757115 = r4757113 / r4757114;
        double r4757116 = r4757115 / r4757107;
        double r4757117 = 6.358019085320933e+22;
        bool r4757118 = r4757096 <= r4757117;
        double r4757119 = -r4757096;
        double r4757120 = r4757107 * r4757100;
        double r4757121 = r4757105 * r4757120;
        double r4757122 = r4757109 - r4757121;
        double r4757123 = sqrt(r4757122);
        double r4757124 = r4757119 - r4757123;
        double r4757125 = r4757107 * r4757114;
        double r4757126 = r4757124 / r4757125;
        double r4757127 = r4757096 / r4757107;
        double r4757128 = r4757101 - r4757127;
        double r4757129 = 1.0;
        double r4757130 = r4757128 * r4757129;
        double r4757131 = r4757118 ? r4757126 : r4757130;
        double r4757132 = r4757104 ? r4757116 : r4757131;
        double r4757133 = r4757098 ? r4757102 : r4757132;
        return r4757133;
}

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

Original33.8
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -0.03099989563658143

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 6.4

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

    if -0.03099989563658143 < b < 5.992812285264993e-289

    1. Initial program 24.4

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

    3. Applied flip--24.4

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

    4. Simplified16.6

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

    5. Simplified16.6

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

    7. Applied associate-/r*16.6

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

    8. Simplified16.7

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

    if 5.992812285264993e-289 < b < 6.358019085320933e+22

    1. Initial program 10.0

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

    if 6.358019085320933e+22 < b

    1. Initial program 33.1

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified6.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))