quadp (p42, positive)

Average Error: 34.0 → 8.3

Time: 18.5s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(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 -2.7e+148)
   (/ (- b) a)
   (if (<= b -7.5e-220)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 2.1e-6)
       (/ (* c -2.0) (+ b (hypot b (sqrt (* c (* a -4.0))))))
       (/ (- c) b)))))

C

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 <= -2.7e+148) {
		tmp = -b / a;
	} else if (b <= -7.5e-220) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 2.1e-6) {
		tmp = (c * -2.0) / (b + hypot(b, sqrt((c * (a * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+148) {
		tmp = -b / a;
	} else if (b <= -7.5e-220) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 2.1e-6) {
		tmp = (c * -2.0) / (b + Math.hypot(b, Math.sqrt((c * (a * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -2.7e+148:
		tmp = -b / a
	elif b <= -7.5e-220:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= 2.1e-6:
		tmp = (c * -2.0) / (b + math.hypot(b, math.sqrt((c * (a * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

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 <= -2.7e+148)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -7.5e-220)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 2.1e-6)
		tmp = Float64(Float64(c * -2.0) / Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.7e+148)
		tmp = -b / a;
	elseif (b <= -7.5e-220)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= 2.1e-6)
		tmp = (c * -2.0) / (b + hypot(b, sqrt((c * (a * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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, -2.7e+148], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -7.5e-220], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-6], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Target

Original	34.0
Target	20.9
Herbie	8.3

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

Derivation

1. Split input into 4 regimes

2. if b < -2.70000000000000019e148

   1. Initial program 61.6

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

   2. Simplified 61.6

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) 1)) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (neg.f64 -1)) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite=> metadata-eval 1)) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) -1) (*.f64 2 a))): 13 points increase in error, 26 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around -inf 1.9

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

   4. Simplified 1.9

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof
      (/.f64 (neg.f64 b) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b)) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 b a))): 0 points increase in error, 0 points decrease in error

   if -2.70000000000000019e148 < b < -7.5000000000000002e-220

   1. Initial program 7.4

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

   if -7.5000000000000002e-220 < b < 2.0999999999999998e-6

   1. Initial program 24.0

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

   2. Simplified 24.1

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) 1)) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (neg.f64 -1)) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite=> metadata-eval 1)) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) -1) (*.f64 2 a))): 13 points increase in error, 26 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 30.4

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

   4. Simplified 30.4

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}{\left(b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -4}\right)\right) \cdot \left(a \cdot -2\right)}} \]
      Proof
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 (*.f64 c a) -4))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 a c)) -4))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 b b (Rewrite<= associate-*r*_binary64 (*.f64 a (*.f64 c -4))))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b b) (*.f64 a (*.f64 c -4))))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a (*.f64 c -4)) (*.f64 b b)))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (Rewrite<= fma-udef_binary64 (fma.f64 a (*.f64 c -4) (*.f64 b b)))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (*.f64 c a) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -4) (*.f64 b b))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 a c)) -4)))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -4) (*.f64 b b))) (*.f64 (+.f64 b (hypot.f64 b (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 a (*.f64 c -4)))))) (*.f64 a -2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 b b) (fma.f64 a (*.f64 c -4) (*.f64 b b))) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 a -2) (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 a (*.f64 c -4)))))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in b around 0 27.4

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

   6. Simplified 27.4

      \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -4}\right)\right) \cdot \left(a \cdot -2\right)} \]
      Proof
      (*.f64 c (*.f64 a 4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c a) 4)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 c a))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 51.4

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \left(-2 \cdot \frac{a}{a}\right)\right)} - 1} \]

   8. Simplified 14.7

      \[\leadsto \color{blue}{\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]
      Proof
      (/.f64 (*.f64 c -2) (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 c (Rewrite<= metadata-eval (*.f64 -2 1))) (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 c (*.f64 -2 (Rewrite<= *-inverses_binary64 (/.f64 a a)))) (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))) (*.f64 -2 (/.f64 a a)))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (/.f64 c (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))) (*.f64 -2 (/.f64 a a)))))): 49 points increase in error, 2 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (/.f64 c (+.f64 b (hypot.f64 b (sqrt.f64 (*.f64 c (*.f64 a -4)))))) (*.f64 -2 (/.f64 a a))))) 1)): 67 points increase in error, 1 points decrease in error

   if 2.0999999999999998e-6 < b

   1. Initial program 55.5

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

   2. Simplified 55.5

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      Proof
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -4))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 4))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) 1)) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 -1/2 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (neg.f64 -1)) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite=> metadata-eval 1)) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (/.f64 (/.f64 -1 2) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) -1) (*.f64 2 a))): 13 points increase in error, 26 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 6.1

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

   4. Simplified 6.1

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 c)) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Error	10.5
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-298}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Error	45.1
Cost	256

\[\frac{-b}{a} \]

Alternative 7
Error	62.3
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2022329 
(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)))