Average Error: 33.7 → 10.6
Time: 26.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3218351 = b;
        double r3218352 = -r3218351;
        double r3218353 = r3218351 * r3218351;
        double r3218354 = 4.0;
        double r3218355 = a;
        double r3218356 = c;
        double r3218357 = r3218355 * r3218356;
        double r3218358 = r3218354 * r3218357;
        double r3218359 = r3218353 - r3218358;
        double r3218360 = sqrt(r3218359);
        double r3218361 = r3218352 - r3218360;
        double r3218362 = 2.0;
        double r3218363 = r3218362 * r3218355;
        double r3218364 = r3218361 / r3218363;
        return r3218364;
}
↓
double f(double a, double b, double c) {
        double r3218365 = b;
        double r3218366 = -7.363255598823911e-15;
        bool r3218367 = r3218365 <= r3218366;
        double r3218368 = c;
        double r3218369 = -r3218368;
        double r3218370 = r3218369 / r3218365;
        double r3218371 = -6.936587154412951e-28;
        bool r3218372 = r3218365 <= r3218371;
        double r3218373 = -r3218365;
        double r3218374 = 2.0;
        double r3218375 = a;
        double r3218376 = r3218374 * r3218375;
        double r3218377 = r3218373 / r3218376;
        double r3218378 = 1.0;
        double r3218379 = r3218378 / r3218376;
        double r3218380 = r3218365 * r3218365;
        double r3218381 = r3218375 * r3218368;
        double r3218382 = 4.0;
        double r3218383 = r3218381 * r3218382;
        double r3218384 = r3218380 - r3218383;
        double r3218385 = sqrt(r3218384);
        double r3218386 = r3218379 * r3218385;
        double r3218387 = r3218377 - r3218386;
        double r3218388 = -2.3344326820285623e-123;
        bool r3218389 = r3218365 <= r3218388;
        double r3218390 = 1.6691257204922504e+85;
        bool r3218391 = r3218365 <= r3218390;
        double r3218392 = r3218376 / r3218385;
        double r3218393 = r3218378 / r3218392;
        double r3218394 = r3218377 - r3218393;
        double r3218395 = r3218368 / r3218365;
        double r3218396 = r3218365 / r3218375;
        double r3218397 = r3218395 - r3218396;
        double r3218398 = r3218391 ? r3218394 : r3218397;
        double r3218399 = r3218389 ? r3218370 : r3218398;
        double r3218400 = r3218372 ? r3218387 : r3218399;
        double r3218401 = r3218367 ? r3218370 : r3218400;
        return r3218401;
}

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

Original33.7
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    3. Applied div-sub51.4

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

    4. Taylor expanded around -inf 10.6

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

    5. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28

    1. Initial program 35.8

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

    3. Applied div-sub35.8

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

    5. Applied div-inv35.9

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

    if -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 12.6

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

    3. Applied div-sub12.6

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

    5. Applied clear-num12.7

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))