Average Error: 34.2 → 10.4
Time: 7.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
(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 -1.5e+86)
   (/ (- b) a)
   (if (<= b 6.2e-102)
     (/ (- (sqrt (fma c (* a -4.0) (* b b))) 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 <= -1.5e+86) {
		tmp = -b / a;
	} else if (b <= 6.2e-102) {
		tmp = (sqrt(fma(c, (a * -4.0), (b * b))) - 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 <= -1.5e+86)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.2e-102)
		tmp = Float64(Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) - 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, -1.5e+86], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.2e-102], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $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 -1.5 \cdot 10^{+86}:\\
\;\;\;\;\frac{-b}{a}\\

\mathbf{elif}\;b \leq 6.2 \cdot 10^{-102}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\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

Original34.2
Target21.0
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 < -1.49999999999999988e86

    1. Initial program 43.9

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

    2. Taylor expanded in b around -inf 4.7

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

    3. Simplified4.7

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

    if -1.49999999999999988e86 < b < 6.20000000000000026e-102

    1. Initial program 12.7

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

    2. Taylor expanded in b around 0 12.7

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

    3. Simplified12.7

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

    if 6.20000000000000026e-102 < b

    1. Initial program 52.1

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

    2. Taylor expanded in b around inf 10.4

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

    3. Simplified10.4

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022172 
(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)))