Average Error: 34.3 → 10.6
Time: 6.8s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+109}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
(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 -1.3e-120)
   (/ (- c) b)
   (if (<= b 3.1e+109)
     (* (+ b (sqrt (fma a (* c -4.0) (* b b)))) (/ -0.5 a))
     (- (/ 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 <= -1.3e-120) {
		tmp = -c / b;
	} else if (b <= 3.1e+109) {
		tmp = (b + sqrt(fma(a, (c * -4.0), (b * b)))) * (-0.5 / a);
	} else {
		tmp = -(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 <= -1.3e-120)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.1e+109)
		tmp = Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) * Float64(-0.5 / a));
	else
		tmp = Float64(-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, -1.3e-120], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.1e+109], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $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 -1.3 \cdot 10^{-120}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \leq 3.1 \cdot 10^{+109}:\\
\;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\

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


\end{array}

Error

Target

Original34.3
Target21.3
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3000000000000001e-120

    1. Initial program 51.0

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

    2. Simplified51.0

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

    3. Taylor expanded in b around -inf 11.5

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

    4. Simplified11.5

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

    if -1.3000000000000001e-120 < b < 3.09999999999999992e109

    1. Initial program 12.1

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

    2. Simplified12.2

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

    if 3.09999999999999992e109 < b

    1. Initial program 49.3

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

    2. Simplified49.4

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

    3. Applied egg-rr38.5

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

    4. Applied egg-rr38.5

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

    5. Taylor expanded in a around 0 3.6

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

    6. Simplified3.6

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+109}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))