Average Error: 34.7 → 8.5
Time: 31.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 123203260287366115360768:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5024024 = b;
        double r5024025 = -r5024024;
        double r5024026 = r5024024 * r5024024;
        double r5024027 = 4.0;
        double r5024028 = a;
        double r5024029 = r5024027 * r5024028;
        double r5024030 = c;
        double r5024031 = r5024029 * r5024030;
        double r5024032 = r5024026 - r5024031;
        double r5024033 = sqrt(r5024032);
        double r5024034 = r5024025 + r5024033;
        double r5024035 = 2.0;
        double r5024036 = r5024035 * r5024028;
        double r5024037 = r5024034 / r5024036;
        return r5024037;
}
↓
double f(double a, double b, double c) {
        double r5024038 = b;
        double r5024039 = -1.2705286994550075e+152;
        bool r5024040 = r5024038 <= r5024039;
        double r5024041 = c;
        double r5024042 = r5024041 / r5024038;
        double r5024043 = a;
        double r5024044 = r5024038 / r5024043;
        double r5024045 = r5024042 - r5024044;
        double r5024046 = 1.0;
        double r5024047 = r5024045 * r5024046;
        double r5024048 = 2.1777947371956335e-143;
        bool r5024049 = r5024038 <= r5024048;
        double r5024050 = r5024038 * r5024038;
        double r5024051 = 4.0;
        double r5024052 = r5024041 * r5024051;
        double r5024053 = r5024052 * r5024043;
        double r5024054 = r5024050 - r5024053;
        double r5024055 = sqrt(r5024054);
        double r5024056 = r5024055 - r5024038;
        double r5024057 = 2.0;
        double r5024058 = r5024043 * r5024057;
        double r5024059 = r5024056 / r5024058;
        double r5024060 = 1.2320326028736612e+23;
        bool r5024061 = r5024038 <= r5024060;
        double r5024062 = r5024050 - r5024050;
        double r5024063 = r5024043 * r5024051;
        double r5024064 = r5024041 * r5024063;
        double r5024065 = r5024062 + r5024064;
        double r5024066 = -r5024038;
        double r5024067 = r5024050 - r5024064;
        double r5024068 = sqrt(r5024067);
        double r5024069 = r5024066 - r5024068;
        double r5024070 = r5024065 / r5024069;
        double r5024071 = r5024070 / r5024058;
        double r5024072 = -1.0;
        double r5024073 = r5024042 * r5024072;
        double r5024074 = r5024061 ? r5024071 : r5024073;
        double r5024075 = r5024049 ? r5024059 : r5024074;
        double r5024076 = r5024040 ? r5024047 : r5024075;
        return r5024076;
}

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

Original34.7
Target21.1
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2705286994550075e+152

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

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

    if -1.2705286994550075e+152 < b < 2.1777947371956335e-143

    1. Initial program 10.4

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

    3. Applied div-inv10.5

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

    5. Applied associate-*r/10.4

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

    6. Simplified10.4

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

    if 2.1777947371956335e-143 < b < 1.2320326028736612e+23

    1. Initial program 36.9

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

    3. Applied flip-+37.0

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

    4. Simplified18.3

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

    if 1.2320326028736612e+23 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 4.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))