Average Error: 34.6 → 9.6
Time: 6.0s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.518100183308216 \cdot 10^{-58}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.543614676518798 \cdot 10^{+140}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
(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 -5.518100183308216e-58)
   (/ (- c) b)
   (if (<= b 1.543614676518798e+140)
     (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) (/ a -0.5))
     (- (/ c b) (/ b a)))))
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 <= -5.518100183308216e-58) {
		tmp = -c / b;
	} else if (b <= 1.543614676518798e+140) {
		tmp = (b + sqrt(fma(a, (c * -4.0), (b * b)))) / (a / -0.5);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end
↓
function code(a, b, c)
	tmp = 0.0
	if (b <= -5.518100183308216e-58)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.543614676518798e+140)
		tmp = Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / Float64(a / -0.5));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := If[LessEqual[b, -5.518100183308216e-58], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.543614676518798e+140], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -5.518100183308216 \cdot 10^{-58}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \leq 1.543614676518798 \cdot 10^{+140}:\\
\;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\frac{a}{-0.5}}\\

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


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target21.2
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.51810018330821617e-58

    1. Initial program 54.6

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

    2. Taylor expanded in b around -inf 7.6

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

    3. Simplified7.6

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

    if -5.51810018330821617e-58 < b < 1.54361467651879794e140

    1. Initial program 13.0

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

    2. Simplified13.1

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

    3. Applied egg-rr13.1

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}}} \]

    4. Applied egg-rr13.0

      \[\leadsto \color{blue}{{\left(\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\frac{a}{-0.5}}\right)}^{1}} \]

    if 1.54361467651879794e140 < b

    1. Initial program 58.8

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

    2. Taylor expanded in b around inf 2.3

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.518100183308216 \cdot 10^{-58}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.543614676518798 \cdot 10^{+140}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :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)))