Average Error: 34.0 → 10.0
Time: 11.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.996635431194612 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7919095286000728 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -3.996635431194612 \cdot 10^{+121}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 2.7919095286000728 \cdot 10^{-58}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}\\

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


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -3.996635431194612e+121)
   (- (/ c b) (/ b a))
   (if (<= b 2.7919095286000728e-58)
     (/ (* (- (sqrt (fma a (* c -4.0) (* b b))) b) 0.5) a)
     (- (/ c b)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.996635431194612e+121) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7919095286000728e-58) {
		tmp = ((sqrt(fma(a, (c * -4.0), (b * b))) - b) * 0.5) / a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.996635431194612e121

    1. Initial program 52.9

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

    2. Taylor expanded in b around -inf 3.5

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

    if -3.996635431194612e121 < b < 2.7919095286000728e-58

    1. Initial program 13.4

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

    2. Simplified13.5

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

    3. Applied egg-rr13.4

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}} \]

    4. Applied egg-rr13.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + \left(-b\right)\right)} \cdot 0.5}{a} \]

    if 2.7919095286000728e-58 < b

    1. Initial program 54.0

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

    2. Taylor expanded in b around inf 8.0

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

    3. Simplified8.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.996635431194612 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7919095286000728 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :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)))