Average Error: 34.2 → 13.8
Time: 20.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\

\mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6990103 = b;
        double r6990104 = -r6990103;
        double r6990105 = r6990103 * r6990103;
        double r6990106 = 4.0;
        double r6990107 = a;
        double r6990108 = r6990106 * r6990107;
        double r6990109 = c;
        double r6990110 = r6990108 * r6990109;
        double r6990111 = r6990105 - r6990110;
        double r6990112 = sqrt(r6990111);
        double r6990113 = r6990104 + r6990112;
        double r6990114 = 2.0;
        double r6990115 = r6990114 * r6990107;
        double r6990116 = r6990113 / r6990115;
        return r6990116;
}
↓
double f(double a, double b, double c) {
        double r6990117 = b;
        double r6990118 = -3.7108875578650606e+138;
        bool r6990119 = r6990117 <= r6990118;
        double r6990120 = 2.0;
        double r6990121 = r6990117 / r6990120;
        double r6990122 = -2.0;
        double r6990123 = r6990121 * r6990122;
        double r6990124 = a;
        double r6990125 = r6990123 / r6990124;
        double r6990126 = 1.627160326743934e-46;
        bool r6990127 = r6990117 <= r6990126;
        double r6990128 = 1.0;
        double r6990129 = sqrt(r6990120);
        double r6990130 = sqrt(r6990129);
        double r6990131 = r6990128 / r6990130;
        double r6990132 = r6990117 * r6990117;
        double r6990133 = 4.0;
        double r6990134 = r6990133 * r6990124;
        double r6990135 = c;
        double r6990136 = r6990134 * r6990135;
        double r6990137 = r6990132 - r6990136;
        double r6990138 = sqrt(r6990137);
        double r6990139 = r6990138 - r6990117;
        double r6990140 = r6990139 / r6990130;
        double r6990141 = r6990140 / r6990129;
        double r6990142 = r6990131 * r6990141;
        double r6990143 = r6990142 / r6990124;
        double r6990144 = -1.0;
        double r6990145 = r6990124 * r6990135;
        double r6990146 = r6990145 / r6990117;
        double r6990147 = r6990144 * r6990146;
        double r6990148 = r6990147 / r6990124;
        double r6990149 = r6990127 ? r6990143 : r6990148;
        double r6990150 = r6990119 ? r6990125 : r6990149;
        return r6990150;
}

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

Original34.2
Target21.0
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    4. Applied add-sqr-sqrt58.6

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

    5. Applied associate-/r*58.6

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

    6. Taylor expanded around -inf 3.3

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

    7. Simplified2.1

      \[\leadsto \frac{\color{blue}{\frac{b}{2} \cdot -2}}{a}\]

    if -3.7108875578650606e+138 < b < 1.627160326743934e-46

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied add-sqr-sqrt13.8

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

    5. Applied associate-/r*13.5

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

    7. Applied *-un-lft-identity13.5

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

    8. Applied sqrt-prod13.5

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

    9. Applied add-sqr-sqrt13.5

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

    10. Applied sqrt-prod13.1

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

    11. Applied *-un-lft-identity13.1

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

    12. Applied times-frac13.2

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

    13. Applied times-frac13.2

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

    14. Simplified13.2

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

    if 1.627160326743934e-46 < b

    1. Initial program 54.2

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

    2. Simplified54.2

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

    3. Taylor expanded around inf 18.5

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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