Average Error: 33.8 → 10.3
Time: 24.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3927293 = b;
        double r3927294 = -r3927293;
        double r3927295 = r3927293 * r3927293;
        double r3927296 = 4.0;
        double r3927297 = a;
        double r3927298 = c;
        double r3927299 = r3927297 * r3927298;
        double r3927300 = r3927296 * r3927299;
        double r3927301 = r3927295 - r3927300;
        double r3927302 = sqrt(r3927301);
        double r3927303 = r3927294 - r3927302;
        double r3927304 = 2.0;
        double r3927305 = r3927304 * r3927297;
        double r3927306 = r3927303 / r3927305;
        return r3927306;
}
↓
double f(double a, double b, double c) {
        double r3927307 = b;
        double r3927308 = -2.5694949190681246e-64;
        bool r3927309 = r3927307 <= r3927308;
        double r3927310 = -1.0;
        double r3927311 = c;
        double r3927312 = r3927311 / r3927307;
        double r3927313 = r3927310 * r3927312;
        double r3927314 = 2.865381670376961e+117;
        bool r3927315 = r3927307 <= r3927314;
        double r3927316 = 1.0;
        double r3927317 = 2.0;
        double r3927318 = a;
        double r3927319 = r3927317 * r3927318;
        double r3927320 = -r3927307;
        double r3927321 = r3927307 * r3927307;
        double r3927322 = 4.0;
        double r3927323 = r3927318 * r3927322;
        double r3927324 = r3927323 * r3927311;
        double r3927325 = r3927321 - r3927324;
        double r3927326 = sqrt(r3927325);
        double r3927327 = r3927320 - r3927326;
        double r3927328 = r3927319 / r3927327;
        double r3927329 = r3927316 / r3927328;
        double r3927330 = r3927307 / r3927318;
        double r3927331 = r3927310 * r3927330;
        double r3927332 = r3927315 ? r3927329 : r3927331;
        double r3927333 = r3927309 ? r3927313 : r3927332;
        return r3927333;
}

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

Original33.8
Target20.9
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.5694949190681246e-64

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.1

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

    if -2.5694949190681246e-64 < b < 2.865381670376961e+117

    1. Initial program 13.1

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    5. Applied clear-num13.2

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

    if 2.865381670376961e+117 < b

    1. Initial program 52.1

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

    2. Taylor expanded around 0 52.1

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

    3. Simplified52.1

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

    5. Applied clear-num52.2

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

    6. Taylor expanded around 0 3.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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