Average Error: 34.0 → 10.4
Time: 10.5s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.593756766636855 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{a}{b} \cdot \frac{-4 \cdot c}{a}\right) \cdot -0.25 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4118194020057405 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
(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 -6.593756766636855e+82)
   (- (* (* (/ a b) (/ (* -4.0 c) a)) -0.25) (/ b a))
   (if (<= b 1.4118194020057405e-46)
     (/ (- (sqrt (fma b b (* -4.0 (* a c)))) b) (* a 2.0))
     (/ (- 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 <= -6.593756766636855e+82) {
		tmp = (((a / b) * ((-4.0 * c) / a)) * -0.25) - (b / a);
	} else if (b <= 1.4118194020057405e-46) {
		tmp = (sqrt(fma(b, b, (-4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	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 <= -6.593756766636855e+82)
		tmp = Float64(Float64(Float64(Float64(a / b) * Float64(Float64(-4.0 * c) / a)) * -0.25) - Float64(b / a));
	elseif (b <= 1.4118194020057405e-46)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	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, -6.593756766636855e+82], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(N[(-4.0 * c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4118194020057405e-46], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $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 -6.593756766636855 \cdot 10^{+82}:\\
\;\;\;\;\left(\frac{a}{b} \cdot \frac{-4 \cdot c}{a}\right) \cdot -0.25 - \frac{b}{a}\\

\mathbf{elif}\;b \leq 1.4118194020057405 \cdot 10^{-46}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\

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


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target21.1
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.59375676663685452e82

    1. Initial program 44.1

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

    2. Simplified44.1

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

    3. Applied egg-rr34.8

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

    4. Taylor expanded in b around -inf 36.0

      \[\leadsto \color{blue}{-\left(0.25 \cdot \frac{{\left(\sqrt{-4 \cdot \left(c \cdot a\right)}\right)}^{2}}{a \cdot b} + \frac{b}{a}\right)} \]

    5. Simplified6.4

      \[\leadsto \color{blue}{\left(\frac{a}{b} \cdot \frac{-4 \cdot c}{a}\right) \cdot -0.25 - \frac{b}{a}} \]

    if -6.59375676663685452e82 < b < 1.4118194020057405e-46

    1. Initial program 14.2

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

    2. Simplified14.3

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

    3. Applied egg-rr19.1

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

    4. Applied egg-rr14.2

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

    if 1.4118194020057405e-46 < b

    1. Initial program 54.3

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

    2. Taylor expanded in b around inf 7.5

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

    3. Simplified7.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.593756766636855 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{a}{b} \cdot \frac{-4 \cdot c}{a}\right) \cdot -0.25 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4118194020057405 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))