quadm (p42, negative)

Average Error: 34.2 → 8.6

Time: 19.0s

Precision: binary64

Cost: 13900

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-272}:\\ \;\;\;\;\left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -1.05e+18)
   (/ (* c -2.0) (fma b 2.0 (* (/ c b) (* -2.0 a))))
   (if (<= b -1.15e-290)
     (/ (* c -2.0) (- b (hypot b (sqrt (* a (* c -4.0))))))
     (if (<= b 1.8e-272)
       (* (+ b (* (sqrt (* a -4.0)) (sqrt c))) (/ -0.5 a))
       (if (<= b 5.8e+81)
         (/ (- (- b) (sqrt (+ (* b b) (* -4.0 (* c a))))) (* 2.0 a))
         (- (/ c b) (/ 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 <= -1.05e+18) {
		tmp = (c * -2.0) / fma(b, 2.0, ((c / b) * (-2.0 * a)));
	} else if (b <= -1.15e-290) {
		tmp = (c * -2.0) / (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else if (b <= 1.8e-272) {
		tmp = (b + (sqrt((a * -4.0)) * sqrt(c))) * (-0.5 / a);
	} else if (b <= 5.8e+81) {
		tmp = (-b - sqrt(((b * b) + (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (c / b) - (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 <= -1.05e+18)
		tmp = Float64(Float64(c * -2.0) / fma(b, 2.0, Float64(Float64(c / b) * Float64(-2.0 * a))));
	elseif (b <= -1.15e-290)
		tmp = Float64(Float64(c * -2.0) / Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	elseif (b <= 1.8e-272)
		tmp = Float64(Float64(b + Float64(sqrt(Float64(a * -4.0)) * sqrt(c))) * Float64(-0.5 / a));
	elseif (b <= 5.8e+81)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / b) - 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, -1.05e+18], N[(N[(c * -2.0), $MachinePrecision] / N[(b * 2.0 + N[(N[(c / b), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.15e-290], N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-272], N[(N[(b + N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+81], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	34.2
Target	21.1
Herbie	8.6

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

Derivation

1. Split input into 5 regimes

2. if b < -1.05e18

   1. Initial program 56.1

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

   2. Simplified 56.1

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      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 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= metadata-eval (/.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))): 8 points increase in error, 23 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<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (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<= sub-neg_binary64 (-.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 57.8

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

   4. Taylor expanded in a around 0 29.0

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

   5. Simplified 29.0

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

   6. Taylor expanded in b around -inf 64.0

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

   7. Simplified 4.6

      \[\leadsto \frac{c \cdot -2}{\color{blue}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}} \]
      Proof
      (fma.f64 b 2 (*.f64 (/.f64 c b) (*.f64 -2 a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 -4)) a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (*.f64 (*.f64 1/2 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4)))) a))): 214 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (*.f64 (*.f64 1/2 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2))) a))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (pow.f64 (sqrt.f64 -4) 2) a))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (*.f64 1/2 (Rewrite<= *-commutative_binary64 (*.f64 a (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (/.f64 c b) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 a (pow.f64 (sqrt.f64 -4) 2)) 1/2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c b) (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) 1/2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) b)) 1/2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) b)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b 2) (*.f64 1/2 (/.f64 (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) b)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 b)) (*.f64 1/2 (/.f64 (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) b)) (*.f64 2 b))): 0 points increase in error, 0 points decrease in error

   if -1.05e18 < b < -1.15e-290

   1. Initial program 27.8

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

   2. Simplified 27.8

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      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 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= metadata-eval (/.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))): 8 points increase in error, 23 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<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (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<= sub-neg_binary64 (-.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 28.3

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

   4. Taylor expanded in a around 0 13.5

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

   5. Simplified 13.5

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

   if -1.15e-290 < b < 1.79999999999999984e-272

   1. Initial program 11.9

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

   2. Simplified 11.9

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      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 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= metadata-eval (/.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))): 8 points increase in error, 23 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<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (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<= sub-neg_binary64 (-.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 11.9

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

   4. Taylor expanded in b around 0 11.9

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

   5. Simplified 11.9

      \[\leadsto \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \cdot \frac{-0.5}{a} \]
      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, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -4 (*.f64 c a))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 33.5

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

   if 1.79999999999999984e-272 < b < 5.7999999999999999e81

   1. Initial program 9.5

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

   if 5.7999999999999999e81 < b

   1. Initial program 44.6

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

   2. Simplified 44.7

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}} \]
      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 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (/.f64 (Rewrite<= metadata-eval (/.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))): 8 points increase in error, 23 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<= distribute-neg-out_binary64 (+.f64 (neg.f64 b) (neg.f64 (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<= sub-neg_binary64 (-.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 4.1

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

   4. Simplified 4.1

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof
      (-.f64 (/.f64 c b) (/.f64 b a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 c b) (neg.f64 (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 c b) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-272}:\\ \;\;\;\;\left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	13900

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \leq -7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-272}:\\ \;\;\;\;\left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	13.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Error	13.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Error	22.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	39.8
Cost	256

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

Alternative 8
Error	62.3
Cost	192

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

Reproduce

herbie shell --seed 2022295 
(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)))