Average Error: 34.5 → 10.5
Time: 10.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r76092 = b;
        double r76093 = -r76092;
        double r76094 = r76092 * r76092;
        double r76095 = 4.0;
        double r76096 = a;
        double r76097 = c;
        double r76098 = r76096 * r76097;
        double r76099 = r76095 * r76098;
        double r76100 = r76094 - r76099;
        double r76101 = sqrt(r76100);
        double r76102 = r76093 - r76101;
        double r76103 = 2.0;
        double r76104 = r76103 * r76096;
        double r76105 = r76102 / r76104;
        return r76105;
}
↓
double f(double a, double b, double c) {
        double r76106 = b;
        double r76107 = -4.7828589349284326e-126;
        bool r76108 = r76106 <= r76107;
        double r76109 = -1.0;
        double r76110 = c;
        double r76111 = r76110 / r76106;
        double r76112 = r76109 * r76111;
        double r76113 = 3.6627135292415903e+111;
        bool r76114 = r76106 <= r76113;
        double r76115 = -r76106;
        double r76116 = r76106 * r76106;
        double r76117 = 4.0;
        double r76118 = a;
        double r76119 = r76118 * r76110;
        double r76120 = r76117 * r76119;
        double r76121 = r76116 - r76120;
        double r76122 = sqrt(r76121);
        double r76123 = r76115 - r76122;
        double r76124 = 2.0;
        double r76125 = r76124 * r76118;
        double r76126 = r76123 / r76125;
        double r76127 = -2.0;
        double r76128 = r76127 * r76106;
        double r76129 = r76128 / r76125;
        double r76130 = r76114 ? r76126 : r76129;
        double r76131 = r76108 ? r76112 : r76130;
        return r76131;
}

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
Target20.6
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b < 3.6627135292415903e+111

    1. Initial program 12.0

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

    if 3.6627135292415903e+111 < b

    1. Initial program 49.7

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

    3. Applied flip--63.3

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.7

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))