Average Error: 34.0 → 9.5
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6487898413435469 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.43504028250552318 \cdot 10^{146}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - 0.5 \cdot \frac{b}{a}\\ \end{array}\]
\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 -2.6487898413435469 \cdot 10^{-64}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.43504028250552318 \cdot 10^{146}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - 0.5 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r87369 = b;
        double r87370 = -r87369;
        double r87371 = r87369 * r87369;
        double r87372 = 4.0;
        double r87373 = a;
        double r87374 = c;
        double r87375 = r87373 * r87374;
        double r87376 = r87372 * r87375;
        double r87377 = r87371 - r87376;
        double r87378 = sqrt(r87377);
        double r87379 = r87370 - r87378;
        double r87380 = 2.0;
        double r87381 = r87380 * r87373;
        double r87382 = r87379 / r87381;
        return r87382;
}
↓
double f(double a, double b, double c) {
        double r87383 = b;
        double r87384 = -2.648789841343547e-64;
        bool r87385 = r87383 <= r87384;
        double r87386 = -1.0;
        double r87387 = c;
        double r87388 = r87387 / r87383;
        double r87389 = r87386 * r87388;
        double r87390 = 1.4350402825055232e+146;
        bool r87391 = r87383 <= r87390;
        double r87392 = -r87383;
        double r87393 = 2.0;
        double r87394 = a;
        double r87395 = r87393 * r87394;
        double r87396 = r87392 / r87395;
        double r87397 = r87383 * r87383;
        double r87398 = 4.0;
        double r87399 = r87394 * r87387;
        double r87400 = r87398 * r87399;
        double r87401 = r87397 - r87400;
        double r87402 = sqrt(r87401);
        double r87403 = r87402 / r87395;
        double r87404 = r87396 - r87403;
        double r87405 = 0.5;
        double r87406 = r87383 / r87394;
        double r87407 = r87405 * r87406;
        double r87408 = r87396 - r87407;
        double r87409 = r87391 ? r87404 : r87408;
        double r87410 = r87385 ? r87389 : r87409;
        return r87410;
}

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

Original34.0
Target20.7
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.648789841343547e-64

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 7.9

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

    if -2.648789841343547e-64 < b < 1.4350402825055232e+146

    1. Initial program 12.4

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

    3. Applied div-sub12.4

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

    if 1.4350402825055232e+146 < b

    1. Initial program 60.9

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

    3. Applied div-sub60.9

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

    5. Applied clear-num60.9

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

    6. Taylor expanded around 0 2.8

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{0.5 \cdot \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6487898413435469 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.43504028250552318 \cdot 10^{146}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - 0.5 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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