The quadratic formula (r2)

Average Error: 33.7 → 11.1

Time: 16.4s

Precision: binary64

Cost: 7952

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 -1.45e-21)
   (/ (- c) b)
   (if (<= b -1.8e-59)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* (* c a) -4.0)))))
     (if (<= b -1.3e-109)
       (/ 0.5 (* 0.5 (- (/ a b) (/ b c))))
       (if (<= b 2e+43)
         (/ (- (- b) (sqrt (+ (* b b) (* c (* a -4.0))))) (* a 2.0))
         (- (/ 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 <= -1.45e-21) {
		tmp = -c / b;
	} else if (b <= -1.8e-59) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + ((c * a) * -4.0))));
	} else if (b <= -1.3e-109) {
		tmp = 0.5 / (0.5 * ((a / b) - (b / c)));
	} else if (b <= 2e+43) {
		tmp = (-b - sqrt(((b * b) + (c * (a * -4.0))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.45d-21)) then
        tmp = -c / b
    else if (b <= (-1.8d-59)) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + ((c * a) * (-4.0d0)))))
    else if (b <= (-1.3d-109)) then
        tmp = 0.5d0 / (0.5d0 * ((a / b) - (b / c)))
    else if (b <= 2d+43) then
        tmp = (-b - sqrt(((b * b) + (c * (a * (-4.0d0)))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

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 <= -1.45e-21) {
		tmp = -c / b;
	} else if (b <= -1.8e-59) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + ((c * a) * -4.0))));
	} else if (b <= -1.3e-109) {
		tmp = 0.5 / (0.5 * ((a / b) - (b / c)));
	} else if (b <= 2e+43) {
		tmp = (-b - Math.sqrt(((b * b) + (c * (a * -4.0))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= -1.45e-21:
		tmp = -c / b
	elif b <= -1.8e-59:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + ((c * a) * -4.0))))
	elif b <= -1.3e-109:
		tmp = 0.5 / (0.5 * ((a / b) - (b / c)))
	elif b <= 2e+43:
		tmp = (-b - math.sqrt(((b * b) + (c * (a * -4.0))))) / (a * 2.0)
	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 <= -1.45e-21)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -1.8e-59)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0)))));
	elseif (b <= -1.3e-109)
		tmp = Float64(0.5 / Float64(0.5 * Float64(Float64(a / b) - Float64(b / c))));
	elseif (b <= 2e+43)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	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 <= -1.45e-21)
		tmp = -c / b;
	elseif (b <= -1.8e-59)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + ((c * a) * -4.0))));
	elseif (b <= -1.3e-109)
		tmp = 0.5 / (0.5 * ((a / b) - (b / c)));
	elseif (b <= 2e+43)
		tmp = (-b - sqrt(((b * b) + (c * (a * -4.0))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	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, -1.45e-21], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -1.8e-59], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.3e-109], N[(0.5 / N[(0.5 * N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+43], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	33.7
Target	20.6
Herbie	11.1

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

Derivation

1. Split input into 5 regimes

2. if b < -1.45e-21

   1. Initial program 54.7

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

   2. Simplified 54.7

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

   3. Taylor expanded in b around -inf 6.7

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

   4. Simplified 6.7

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_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

   if -1.45e-21 < b < -1.8e-59

   1. Initial program 37.2

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

   2. Simplified 37.2

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 b b))))): 0 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 (*.f64 a c) (neg.f64 4)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_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, 0 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))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite=> remove-double-neg_binary64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a))): 9 points increase in error, 25 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 37.2

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

   if -1.8e-59 < b < -1.2999999999999999e-109

   1. Initial program 28.9

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

   2. Simplified 29.0

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 b b))))): 0 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 (*.f64 a c) (neg.f64 4)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_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, 0 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))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite=> remove-double-neg_binary64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a))): 9 points increase in error, 25 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 29.7

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

   4. Taylor expanded in b around -inf 64.0

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

   5. Simplified 38.2

      \[\leadsto \frac{-0.5}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.5 \cdot \frac{b}{c}\right)}} \]
      Proof
      (fma.f64 -1/2 (/.f64 a b) (*.f64 1/2 (/.f64 b c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (*.f64 (Rewrite<= metadata-eval (/.f64 -2 -4)) (/.f64 b c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (*.f64 (/.f64 -2 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4)))) (/.f64 b c))): 182 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (*.f64 (/.f64 -2 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2))) (/.f64 b c))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -2 b) (*.f64 (pow.f64 (sqrt.f64 -4) 2) c)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (/.f64 (*.f64 -2 b) (Rewrite<= *-commutative_binary64 (*.f64 c (pow.f64 (sqrt.f64 -4) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 a b) (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 a b)) (*.f64 -2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -2 (/.f64 b (*.f64 c (pow.f64 (sqrt.f64 -4) 2)))) (*.f64 -1/2 (/.f64 a b)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 38.2

      \[\leadsto \color{blue}{0 + \frac{-0.5}{\mathsf{fma}\left(0.5, \frac{b}{c}, \frac{-0.5 \cdot a}{b}\right)}} \]

   7. Simplified 38.2

      \[\leadsto \color{blue}{\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}} \]
      Proof
      (/.f64 1/2 (*.f64 1/2 (-.f64 (/.f64 a b) (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 1/2 (/.f64 a b)) (*.f64 1/2 (/.f64 b c))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (-.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1/2)) (/.f64 a b)) (*.f64 1/2 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 -1/2 (/.f64 a b)))) (*.f64 1/2 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (-.f64 (neg.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1/2 a) b))) (*.f64 1/2 (/.f64 b c)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (/.f64 (*.f64 -1/2 a) b)) (neg.f64 (*.f64 1/2 (/.f64 b c)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (/.f64 (*.f64 -1/2 a) b) (*.f64 1/2 (/.f64 b c)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 b c)) (/.f64 (*.f64 -1/2 a) b))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (neg.f64 (Rewrite<= fma-udef_binary64 (fma.f64 1/2 (/.f64 b c) (/.f64 (*.f64 -1/2 a) b))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/2 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (fma.f64 1/2 (/.f64 b c) (/.f64 (*.f64 -1/2 a) b))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1/2 -1) (fma.f64 1/2 (/.f64 b c) (/.f64 (*.f64 -1/2 a) b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> metadata-eval -1/2) (fma.f64 1/2 (/.f64 b c) (/.f64 (*.f64 -1/2 a) b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 -1/2 (fma.f64 1/2 (/.f64 b c) (/.f64 (*.f64 -1/2 a) b))))): 0 points increase in error, 0 points decrease in error

   if -1.2999999999999999e-109 < b < 2.00000000000000003e43

   1. Initial program 13.2

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

   2. Simplified 13.2

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

   if 2.00000000000000003e43 < b

   1. Initial program 37.5

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

   2. Simplified 37.6

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)} \]
      Proof
      (*.f64 (/.f64 -1/2 a) (+.f64 b (sqrt.f64 (fma.f64 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) -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 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 b b))))): 0 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 (*.f64 a c) (neg.f64 4)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_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, 0 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))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (Rewrite=> sub-neg_binary64 (+.f64 b (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1 (*.f64 2 a)) (+.f64 b (Rewrite=> remove-double-neg_binary64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c)))))) (*.f64 2 a))): 9 points increase in error, 25 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 a around 0 6.3

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

   4. Simplified 6.3

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      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 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_0\right)\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	11.0
Cost	7888

\[\begin{array}{l} t_0 := \frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	14.3
Cost	7632

\[\begin{array}{l} t_0 := -0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-109}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	14.4
Cost	7632

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{\left(c \cdot a\right) \cdot -4}}}\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{0.5}{0.5 \cdot \left(\frac{a}{b} - \frac{b}{c}\right)}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Error	39.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-15}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-271}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Error	62.3
Cost	192

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

Alternative 9
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(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)))