Average Error: 33.6 → 11.3
Time: 24.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)\right) \cdot \frac{\frac{1}{2}}{a}\\ \end{array}\]
\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 -1.2890050783826923 \cdot 10^{-183}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)\right) \cdot \frac{\frac{1}{2}}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r1415914 = b;
        double r1415915 = -r1415914;
        double r1415916 = r1415914 * r1415914;
        double r1415917 = 4.0;
        double r1415918 = a;
        double r1415919 = c;
        double r1415920 = r1415918 * r1415919;
        double r1415921 = r1415917 * r1415920;
        double r1415922 = r1415916 - r1415921;
        double r1415923 = sqrt(r1415922);
        double r1415924 = r1415915 - r1415923;
        double r1415925 = 2.0;
        double r1415926 = r1415925 * r1415918;
        double r1415927 = r1415924 / r1415926;
        return r1415927;
}
↓
double f(double a, double b, double c) {
        double r1415928 = b;
        double r1415929 = -1.2890050783826923e-183;
        bool r1415930 = r1415928 <= r1415929;
        double r1415931 = c;
        double r1415932 = r1415931 / r1415928;
        double r1415933 = -r1415932;
        double r1415934 = 1.786204067849289e+100;
        bool r1415935 = r1415928 <= r1415934;
        double r1415936 = -r1415928;
        double r1415937 = r1415928 * r1415928;
        double r1415938 = a;
        double r1415939 = r1415931 * r1415938;
        double r1415940 = 4.0;
        double r1415941 = r1415939 * r1415940;
        double r1415942 = r1415937 - r1415941;
        double r1415943 = sqrt(r1415942);
        double r1415944 = r1415936 - r1415943;
        double r1415945 = 0.5;
        double r1415946 = r1415945 / r1415938;
        double r1415947 = r1415944 * r1415946;
        double r1415948 = 2.0;
        double r1415949 = r1415928 / r1415931;
        double r1415950 = r1415938 / r1415949;
        double r1415951 = r1415950 - r1415928;
        double r1415952 = r1415948 * r1415951;
        double r1415953 = r1415952 * r1415946;
        double r1415954 = r1415935 ? r1415947 : r1415953;
        double r1415955 = r1415930 ? r1415933 : r1415954;
        return r1415955;
}

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

Original33.6
Target20.5
Herbie11.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2890050783826923e-183

    1. Initial program 48.2

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

    3. Applied clear-num48.2

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

    4. Simplified48.2

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

    5. Taylor expanded around -inf 14.3

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

    6. Simplified14.3

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

    if -1.2890050783826923e-183 < b < 1.786204067849289e+100

    1. Initial program 10.5

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

    3. Applied div-inv10.7

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

    4. Simplified10.7

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

    if 1.786204067849289e+100 < b

    1. Initial program 44.2

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

    3. Applied div-inv44.3

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

    4. Simplified44.3

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

    5. Taylor expanded around inf 9.5

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

    6. Simplified3.7

      \[\leadsto \color{blue}{\left(\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2\right)} \cdot \frac{\frac{1}{2}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)\right) \cdot \frac{\frac{1}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))