Average Error: 33.7 → 7.0
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.5975400610846271 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.49040980815040939 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{\frac{-b}{4} \cdot \frac{2}{c} + \frac{-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4} \cdot \frac{2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -9.5975400610846271 \cdot 10^{115}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -5.49040980815040939 \cdot 10^{-305}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\
\;\;\;\;\frac{1}{\frac{-b}{4} \cdot \frac{2}{c} + \frac{-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4} \cdot \frac{2}{c}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r175110 = b;
        double r175111 = -r175110;
        double r175112 = r175110 * r175110;
        double r175113 = 4.0;
        double r175114 = a;
        double r175115 = r175113 * r175114;
        double r175116 = c;
        double r175117 = r175115 * r175116;
        double r175118 = r175112 - r175117;
        double r175119 = sqrt(r175118);
        double r175120 = r175111 + r175119;
        double r175121 = 2.0;
        double r175122 = r175121 * r175114;
        double r175123 = r175120 / r175122;
        return r175123;
}
↓
double f(double a, double b, double c) {
        double r175124 = b;
        double r175125 = -9.597540061084627e+115;
        bool r175126 = r175124 <= r175125;
        double r175127 = 1.0;
        double r175128 = c;
        double r175129 = r175128 / r175124;
        double r175130 = a;
        double r175131 = r175124 / r175130;
        double r175132 = r175129 - r175131;
        double r175133 = r175127 * r175132;
        double r175134 = -5.490409808150409e-305;
        bool r175135 = r175124 <= r175134;
        double r175136 = -r175124;
        double r175137 = r175124 * r175124;
        double r175138 = 4.0;
        double r175139 = r175138 * r175130;
        double r175140 = r175139 * r175128;
        double r175141 = r175137 - r175140;
        double r175142 = sqrt(r175141);
        double r175143 = r175136 + r175142;
        double r175144 = 2.0;
        double r175145 = r175144 * r175130;
        double r175146 = r175143 / r175145;
        double r175147 = 3.29571176074688e+130;
        bool r175148 = r175124 <= r175147;
        double r175149 = 1.0;
        double r175150 = r175136 / r175138;
        double r175151 = r175144 / r175128;
        double r175152 = r175150 * r175151;
        double r175153 = -r175142;
        double r175154 = r175153 / r175138;
        double r175155 = r175154 * r175151;
        double r175156 = r175152 + r175155;
        double r175157 = r175149 / r175156;
        double r175158 = -1.0;
        double r175159 = r175158 * r175129;
        double r175160 = r175148 ? r175157 : r175159;
        double r175161 = r175135 ? r175146 : r175160;
        double r175162 = r175126 ? r175133 : r175161;
        return r175162;
}

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

Original33.7
Target20.9
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.597540061084627e+115

    1. Initial program 48.7

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -9.597540061084627e+115 < b < -5.490409808150409e-305

    1. Initial program 8.7

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

    if -5.490409808150409e-305 < b < 3.29571176074688e+130

    1. Initial program 34.3

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

    3. Applied flip-+34.4

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

    4. Simplified16.9

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

    6. Applied clear-num17.1

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified16.0

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

    9. Applied associate-/l*16.0

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

    10. Simplified9.7

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

    12. Applied sub-neg9.7

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

    13. Applied distribute-lft-in9.7

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

    14. Simplified9.7

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

    15. Simplified9.7

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

    if 3.29571176074688e+130 < b

    1. Initial program 61.2

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

    2. Taylor expanded around inf 2.1

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.5975400610846271 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.49040980815040939 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{\frac{-b}{4} \cdot \frac{2}{c} + \frac{-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4} \cdot \frac{2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))