The quadratic formula (r2)

Average Error: 34.2 → 10.1

Time: 17.6s

Precision: binary64

Cost: 14152

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{b}{a \cdot -2} + -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -5.6e-74)
   (/ (- c) b)
   (if (<= b 9.5e+124)
     (+ (/ b (* a -2.0)) (* -0.5 (/ (sqrt (fma c (* a -4.0) (* b b))) a)))
     (/ (- b) a))))

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 <= -5.6e-74) {
		tmp = -c / b;
	} else if (b <= 9.5e+124) {
		tmp = (b / (a * -2.0)) + (-0.5 * (sqrt(fma(c, (a * -4.0), (b * b))) / a));
	} else {
		tmp = -b / a;
	}
	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 <= -5.6e-74)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 9.5e+124)
		tmp = Float64(Float64(b / Float64(a * -2.0)) + Float64(-0.5 * Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return 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, -5.6e-74], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 9.5e+124], N[(N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	34.2
Target	21.0
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if b < -5.59999999999999976e-74

   1. Initial program 53.1

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

   2. Simplified 53.1

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 4) a)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 a))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (neg.f64 (*.f64 4 a))) (*.f64 b b)))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 a)) c)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (*.f64 4 a) c))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 4 (*.f64 a c)))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (neg.f64 (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 18 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 -1 (*.f64 2 a)) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 -1 (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (neg.f64 -1) (-.f64 (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
      (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 -1) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 (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
      (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 (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 9.1

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

   4. Simplified 9.1

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 c b))): 0 points increase in error, 3 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 c b))): 3 points increase in error, 0 points decrease in error

   if -5.59999999999999976e-74 < b < 9.50000000000000004e124

   1. Initial program 12.9

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

   2. Simplified 13.1

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 4) a)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 a))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (neg.f64 (*.f64 4 a))) (*.f64 b b)))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 a)) c)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (*.f64 4 a) c))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 4 (*.f64 a c)))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (neg.f64 (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 18 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 -1 (*.f64 2 a)) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 -1 (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (neg.f64 -1) (-.f64 (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
      (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 -1) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 (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
      (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 (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 13.1

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

   4. Applied egg-rr 13.2

      \[\leadsto \frac{-0.5}{\frac{a}{b + \color{blue}{{\left({\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}}}} \]

   5. Applied egg-rr 13.0

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

   6. Simplified 13.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{a}, \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, \frac{b}{a} \cdot -0.5\right)} \]
      Proof
      (fma.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))) (*.f64 (/.f64 b a) -1/2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))) (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 b a)))): 2 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) (*.f64 -1/2 (/.f64 b a)))): 0 points increase in error, 2 points decrease in error
      (+.f64 (*.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 b a) -1/2))): 6 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) (Rewrite<= associate-/r/_binary64 (/.f64 b (/.f64 a -1/2)))): 0 points increase in error, 6 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 b (/.f64 a -1/2)) (*.f64 (/.f64 -1/2 a) (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))))): 6 points increase in error, 0 points decrease in error

   7. Applied egg-rr 12.9

      \[\leadsto \color{blue}{\frac{b}{a \cdot -2} + -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]

   if 9.50000000000000004e124 < b

   1. Initial program 54.1

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

   2. Simplified 54.1

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a))) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 4) a)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 a))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (neg.f64 (*.f64 4 a))) (*.f64 b b)))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 a)) c)) (*.f64 b b))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (*.f64 4 a) c))) (*.f64 b b))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 4 (*.f64 a c)))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (neg.f64 (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 18 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 -1 (*.f64 2 a)) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 -1 (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 -1) (*.f64 2 a))) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))): 0 points increase in error, 22 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (neg.f64 -1) (-.f64 (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
      (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 -1) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 2 a)))): 22 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 (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
      (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 (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 a around 0 3.0

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

   4. Simplified 3.0

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 c b))): 0 points increase in error, 3 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 c b))): 3 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{b}{a \cdot -2} + -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-87}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-123}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-70}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	39.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	22.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	62.3
Cost	192

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

Alternative 7
Error	56.8
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2022343 
(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)))