Average Error: 33.4 → 16.7
Time: 26.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 8.670930634061063 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]
\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 8.670930634061063 \cdot 10^{-143}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2517506 = b;
        double r2517507 = -r2517506;
        double r2517508 = r2517506 * r2517506;
        double r2517509 = 4.0;
        double r2517510 = a;
        double r2517511 = c;
        double r2517512 = r2517510 * r2517511;
        double r2517513 = r2517509 * r2517512;
        double r2517514 = r2517508 - r2517513;
        double r2517515 = sqrt(r2517514);
        double r2517516 = r2517507 + r2517515;
        double r2517517 = 2.0;
        double r2517518 = r2517517 * r2517510;
        double r2517519 = r2517516 / r2517518;
        return r2517519;
}
↓
double f(double a, double b, double c) {
        double r2517520 = b;
        double r2517521 = 8.670930634061063e-143;
        bool r2517522 = r2517520 <= r2517521;
        double r2517523 = a;
        double r2517524 = -4.0;
        double r2517525 = c;
        double r2517526 = r2517524 * r2517525;
        double r2517527 = r2517520 * r2517520;
        double r2517528 = fma(r2517523, r2517526, r2517527);
        double r2517529 = sqrt(r2517528);
        double r2517530 = r2517529 - r2517520;
        double r2517531 = 2.0;
        double r2517532 = r2517530 / r2517531;
        double r2517533 = r2517532 / r2517523;
        double r2517534 = 1.0;
        double r2517535 = r2517520 / r2517525;
        double r2517536 = -r2517535;
        double r2517537 = r2517534 / r2517536;
        double r2517538 = r2517522 ? r2517533 : r2517537;
        return r2517538;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target19.8
Herbie16.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 8.670930634061063e-143

    1. Initial program 20.2

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

    2. Simplified20.2

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

    3. Taylor expanded around 0 20.2

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

    4. Simplified20.2

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

    if 8.670930634061063e-143 < b

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around 0 49.8

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

    4. Simplified49.9

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

    6. Applied *-un-lft-identity49.9

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

    7. Applied associate-/l*49.9

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

    8. Taylor expanded around 0 12.3

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

    9. Simplified12.3

      \[\leadsto \frac{1}{\color{blue}{-\frac{b}{c}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 8.670930634061063 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))