Quadratic roots, full range

Percentage Accurate: 52.9% → 85.7%

Time: 14.4s

Alternatives: 9

Speedup: 11.6×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

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

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

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

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+124)
   (/ (- 0.0 b) a)
   (if (<= b 8e-93)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (/ (/ 0.5 (+ (/ a (* b b)) (/ -1.0 c))) (* b 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+124) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8e-93) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	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
    real(8) :: tmp
    if (b <= (-1.6d+124)) then
        tmp = (0.0d0 - b) / a
    else if (b <= 8d-93) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = (0.5d0 / ((a / (b * b)) + ((-1.0d0) / c))) / (b * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+124) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8e-93) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e+124:
		tmp = (0.0 - b) / a
	elif b <= 8e-93:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+124)
		tmp = Float64(Float64(0.0 - b) / a);
	elseif (b <= 8e-93)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a / Float64(b * b)) + Float64(-1.0 / c))) / Float64(b * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e+124)
		tmp = (0.0 - b) / a;
	elseif (b <= 8e-93)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	else
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+124], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8e-93], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.59999999999999996e124

   1. Initial program 44.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 44.4%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 96.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 96.3%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 96.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 96.3%

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

   if -1.59999999999999996e124 < b < 7.9999999999999992e-93

   1. Initial program 85.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 85.0%

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

   if 7.9999999999999992e-93 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{a}{b \cdot b} - \frac{1}{c}\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}}{\color{blue}{b \cdot \frac{1}{2}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}\right), \color{blue}{\left(b \cdot \frac{1}{2}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{2}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} + \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\frac{a}{b \cdot b}\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-1}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{b}\right)\right) \]

       13. *-lowering-*.f64 82.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{0.5 \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.7e+118)
   (/ (- 0.0 b) a)
   (if (<= b 8.5e-93)
     (/ 0.5 (/ a (- (sqrt (+ (* b b) (* a (* c -4.0)))) b)))
     (/ (/ 0.5 (+ (/ a (* b b)) (/ -1.0 c))) (* b 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e+118) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	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
    real(8) :: tmp
    if (b <= (-5.7d+118)) then
        tmp = (0.0d0 - b) / a
    else if (b <= 8.5d-93) then
        tmp = 0.5d0 / (a / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
    else
        tmp = (0.5d0 / ((a / (b * b)) + ((-1.0d0) / c))) / (b * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.7e+118) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = 0.5 / (a / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.7e+118:
		tmp = (0.0 - b) / a
	elif b <= 8.5e-93:
		tmp = 0.5 / (a / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
	else:
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.7e+118)
		tmp = Float64(Float64(0.0 - b) / a);
	elseif (b <= 8.5e-93)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a / Float64(b * b)) + Float64(-1.0 / c))) / Float64(b * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.7e+118)
		tmp = (0.0 - b) / a;
	elseif (b <= 8.5e-93)
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	else
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.7e+118], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8.5e-93], N[(0.5 / N[(a / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.70000000000000002e118

   1. Initial program 46.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 46.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 96.4%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 96.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 96.4%

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

   if -5.70000000000000002e118 < b < 8.5000000000000007e-93

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 84.5%

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

   if 8.5000000000000007e-93 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{a}{b \cdot b} - \frac{1}{c}\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}}{\color{blue}{b \cdot \frac{1}{2}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}\right), \color{blue}{\left(b \cdot \frac{1}{2}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{2}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} + \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\frac{a}{b \cdot b}\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-1}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{b}\right)\right) \]

       13. *-lowering-*.f64 82.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{0.5 \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e+118)
   (/ (- 0.0 b) a)
   (if (<= b 4.8e-93)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (/ (/ 0.5 (+ (/ a (* b b)) (/ -1.0 c))) (* b 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e+118) {
		tmp = (0.0 - b) / a;
	} else if (b <= 4.8e-93) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	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
    real(8) :: tmp
    if (b <= (-5.4d+118)) then
        tmp = (0.0d0 - b) / a
    else if (b <= 4.8d-93) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = (0.5d0 / ((a / (b * b)) + ((-1.0d0) / c))) / (b * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e+118) {
		tmp = (0.0 - b) / a;
	} else if (b <= 4.8e-93) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e+118:
		tmp = (0.0 - b) / a
	elif b <= 4.8e-93:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e+118)
		tmp = Float64(Float64(0.0 - b) / a);
	elseif (b <= 4.8e-93)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a / Float64(b * b)) + Float64(-1.0 / c))) / Float64(b * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e+118)
		tmp = (0.0 - b) / a;
	elseif (b <= 4.8e-93)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e+118], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.8e-93], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.4e118

   1. Initial program 46.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 46.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 96.4%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 96.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 96.4%

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

   if -5.4e118 < b < 4.8000000000000002e-93

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      13. *-lowering-*.f64 84.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right) \]

   6. Applied egg-rr 84.4%

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

   if 4.8000000000000002e-93 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{a}{b \cdot b} - \frac{1}{c}\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}}{\color{blue}{b \cdot \frac{1}{2}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}\right), \color{blue}{\left(b \cdot \frac{1}{2}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{2}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} + \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\frac{a}{b \cdot b}\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-1}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{b}\right)\right) \]

       13. *-lowering-*.f64 82.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{0.5 \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-86)
   (/ (- 0.0 b) a)
   (if (<= b 8.5e-93)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (/ 0.5 (+ (/ a (* b b)) (/ -1.0 c))) (* b 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-86) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	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
    real(8) :: tmp
    if (b <= (-5.5d-86)) then
        tmp = (0.0d0 - b) / a
    else if (b <= 8.5d-93) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (0.5d0 / ((a / (b * b)) + ((-1.0d0) / c))) / (b * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-86) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-86:
		tmp = (0.0 - b) / a
	elif b <= 8.5e-93:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-86)
		tmp = Float64(Float64(0.0 - b) / a);
	elseif (b <= 8.5e-93)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a / Float64(b * b)) + Float64(-1.0 / c))) / Float64(b * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e-86)
		tmp = (0.0 - b) / a;
	elseif (b <= 8.5e-93)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-86], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8.5e-93], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5e-86

   1. Initial program 66.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 86.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 86.5%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 86.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 86.5%

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

   if -5.5e-86 < b < 8.5000000000000007e-93

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 80.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-4 \cdot a\right) \cdot c\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 77.6%

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

   if 8.5000000000000007e-93 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{a}{b \cdot b} - \frac{1}{c}\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}}{\color{blue}{b \cdot \frac{1}{2}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}\right), \color{blue}{\left(b \cdot \frac{1}{2}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{2}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} + \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\frac{a}{b \cdot b}\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-1}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{b}\right)\right) \]

       13. *-lowering-*.f64 82.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{0.5 \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e-85)
   (/ (- 0.0 b) a)
   (if (<= b 8.5e-93)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (/ (/ 0.5 (+ (/ a (* b b)) (/ -1.0 c))) (* b 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-85) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	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
    real(8) :: tmp
    if (b <= (-1.4d-85)) then
        tmp = (0.0d0 - b) / a
    else if (b <= 8.5d-93) then
        tmp = (0.5d0 / a) * (sqrt((a * (c * (-4.0d0)))) - b)
    else
        tmp = (0.5d0 / ((a / (b * b)) + ((-1.0d0) / c))) / (b * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-85) {
		tmp = (0.0 - b) / a;
	} else if (b <= 8.5e-93) {
		tmp = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e-85:
		tmp = (0.0 - b) / a
	elif b <= 8.5e-93:
		tmp = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	else:
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e-85)
		tmp = Float64(Float64(0.0 - b) / a);
	elseif (b <= 8.5e-93)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a / Float64(b * b)) + Float64(-1.0 / c))) / Float64(b * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.4e-85)
		tmp = (0.0 - b) / a;
	elseif (b <= 8.5e-93)
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	else
		tmp = (0.5 / ((a / (b * b)) + (-1.0 / c))) / (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e-85], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8.5e-93], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.40000000000000008e-85

   1. Initial program 66.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 86.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 86.5%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 86.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 86.5%

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

   if -1.40000000000000008e-85 < b < 8.5000000000000007e-93

   1. Initial program 80.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 80.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-4 \cdot a\right) \cdot c\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 77.6%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}} - b}{a \cdot 2} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{1}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{a \cdot 2} \cdot \color{blue}{\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{a \cdot 2}\right), \left(\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}} - b\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{2 \cdot a}\right), \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{2}}{a}\right), \left(\color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}} - b\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\sqrt{\color{blue}{c} \cdot \left(a \cdot -4\right)} - b\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}} - b\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}} - b\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{c \cdot \left(a \cdot -4\right)}\right), \color{blue}{b}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{\left(a \cdot -4\right) \cdot c}\right), b\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{a \cdot \left(-4 \cdot c\right)}\right), b\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{a \cdot \left(c \cdot -4\right)}\right), b\right)\right) \]

      14. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right)\right) \]

      16. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right)\right) \]

   9. Applied egg-rr 77.3%

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

   if 8.5000000000000007e-93 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

         \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 17.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\frac{a}{b \cdot b} - \frac{1}{c}\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}}{\color{blue}{b \cdot \frac{1}{2}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a}{b \cdot b} - \frac{1}{c}}\right), \color{blue}{\left(b \cdot \frac{1}{2}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{2}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{b \cdot b} + \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\frac{a}{b \cdot b}\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\mathsf{neg}\left(\frac{1}{c}\right)\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-1}{c}\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(b \cdot \frac{1}{2}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{b}\right)\right) \]

       13. *-lowering-*.f64 82.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(-1, c\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{0.5 \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{b \cdot b} + \frac{-1}{c}}}{b \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-303}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7e-303) (/ (- 0.0 b) a) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e-303) {
		tmp = (0.0 - b) / a;
	} else {
		tmp = 0.0 - (c / b);
	}
	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
    real(8) :: tmp
    if (b <= 7d-303) then
        tmp = (0.0d0 - b) / a
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e-303) {
		tmp = (0.0 - b) / a;
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7e-303:
		tmp = (0.0 - b) / a
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7e-303)
		tmp = Float64(Float64(0.0 - b) / a);
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7e-303)
		tmp = (0.0 - b) / a;
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7e-303], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e-303

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 72.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 65.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 65.2%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 65.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 65.2%

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

   if 7e-303 < b

   1. Initial program 33.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 64.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 64.8%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-303}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- 0.0 b) a) (* c (/ -1.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (0.0 - b) / a;
	} else {
		tmp = c * (-1.0 / b);
	}
	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
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (0.0d0 - b) / a
    else
        tmp = c * ((-1.0d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (0.0 - b) / a;
	} else {
		tmp = c * (-1.0 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (0.0 - b) / a
	else:
		tmp = c * (-1.0 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(0.0 - b) / a);
	else
		tmp = Float64(c * Float64(-1.0 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (0.0 - b) / a;
	else
		tmp = c * (-1.0 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 72.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 65.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 65.2%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 65.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 65.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 33.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto c \cdot \left(\frac{-1 \cdot \left(a \cdot c\right)}{{b}^{3}} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{b}}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto c \cdot \left(\frac{\left(-1 \cdot a\right) \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{\color{blue}{1}}{b}\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto c \cdot \left(\frac{-1 \cdot a}{{b}^{3}} \cdot c + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{b}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto c \cdot \left(\left(-1 \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{\color{blue}{1}}{b}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\left(-1 \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right) + \color{blue}{\left(-1 \cdot \frac{a}{{b}^{3}}\right) \cdot c}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right), \color{blue}{\left(\left(-1 \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{b}\right), \left(\color{blue}{\left(-1 \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{-1}{b}\right), \left(\left(\color{blue}{-1} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \left(\color{blue}{\left(-1 \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \left(\frac{-1 \cdot a}{{b}^{3}} \cdot c\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \left(\frac{\left(-1 \cdot a\right) \cdot c}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \left(\frac{-1 \cdot \left(a \cdot c\right)}{{\color{blue}{b}}^{3}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\left(-1 \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\left(-1 \cdot \left(c \cdot a\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\left(\left(-1 \cdot c\right) \cdot a\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot c\right), a\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(c\right)\right), a\right), \left({b}^{3}\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(0 - c\right), a\right), \left({b}^{3}\right)\right)\right)\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), a\right), \left({b}^{3}\right)\right)\right)\right) \]

      22. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), a\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{b} + \frac{\left(0 - c\right) \cdot a}{b \cdot \left(b \cdot b\right)}\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-1}{b}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 64.6%

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(-1, \color{blue}{b}\right)\right) \]

   10. Simplified 64.6%

       \[\leadsto c \cdot \color{blue}{\frac{-1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{0 - b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- 0.0 b) a))

C

double code(double a, double b, double c) {
	return (0.0 - b) / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.0d0 - b) / a
end function

Java

public static double code(double a, double b, double c) {
	return (0.0 - b) / a;
}

Python

def code(a, b, c):
	return (0.0 - b) / a

Julia

function code(a, b, c)
	return Float64(Float64(0.0 - b) / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.0 - b) / a;
end

Wolfram

code[a_, b_, c_] := N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 52.6%

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

5. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

   4. /-lowering-/.f64 34.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

7. Simplified 34.2%

   \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

   3. /-lowering-/.f64 34.2%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

9. Applied egg-rr 34.2%

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

10. Final simplification 34.2%

    \[\leadsto \frac{0 - b}{a} \]
11. Add Preprocessing

Alternative 9: 2.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

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 / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 52.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

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

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}}{2} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{a}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

   13. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

6. Applied egg-rr 52.5%

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

7. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{1}{2} \cdot \frac{1}{c}\right)\right)}\right) \]
8. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(b \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\left(\frac{a}{{b}^{2}} - \frac{1}{c}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(b \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \left(\color{blue}{\frac{a}{{b}^{2}}} - \frac{1}{c}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\frac{a}{{b}^{2}}\right), \color{blue}{\left(\frac{1}{c}\right)}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left({b}^{2}\right)\right), \left(\frac{\color{blue}{1}}{c}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \left(b \cdot b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{1}{c}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 32.7%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(1, \color{blue}{c}\right)\right)\right)\right) \]

9. Simplified 32.7%

   \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot 0.5\right) \cdot \left(\frac{a}{b \cdot b} - \frac{1}{c}\right)}} \]

10. Taylor expanded in b around 0

    \[\leadsto \color{blue}{\frac{b}{a}} \]
11. Step-by-step derivation

    1. /-lowering-/.f64 2.6%

       \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{a}\right) \]

12. Simplified 2.6%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))