quadp (p42, positive)

Percentage Accurate: 52.1% → 83.9%

Time: 14.1s

Alternatives: 13

Speedup: 11.6×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+121)
   (- 0.0 (/ b a))
   (if (<= b 5.5e+31)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+121) {
		tmp = 0.0 - (b / a);
	} else if (b <= 5.5e+31) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-2.2d+121)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 5.5d+31) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+121) {
		tmp = 0.0 - (b / a);
	} else if (b <= 5.5e+31) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e+121:
		tmp = 0.0 - (b / a)
	elif b <= 5.5e+31:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+121)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 5.5e+31)
		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(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e+121)
		tmp = 0.0 - (b / a);
	elseif (b <= 5.5e+31)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+121], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e+31], 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[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.20000000000000001e121

   1. Initial program 67.6%

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

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

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

   3. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 96.7%

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

   7. Simplified 96.7%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 96.7%

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

   9. Applied egg-rr 96.7%

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

   if -2.20000000000000001e121 < b < 5.50000000000000002e31

   1. Initial program 72.7%

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

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

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

   3. Simplified 72.7%

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

   if 5.50000000000000002e31 < b

   1. Initial program 10.4%

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

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

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

   3. Simplified 10.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 10.3%

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

   6. Applied egg-rr 10.3%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 89.2%

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

   9. Simplified 89.2%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 90.2%

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

   11. Applied egg-rr 90.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+121}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{+112}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.05e+112)
   (- 0.0 (/ b a))
   (if (<= b 1.6e+27)
     (/ 0.5 (/ a (- (sqrt (+ (* b b) (* a (* c -4.0)))) b)))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.05e+112) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-3.05d+112)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 1.6d+27) then
        tmp = 0.5d0 / (a / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.05e+112) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = 0.5 / (a / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.05e+112:
		tmp = 0.0 - (b / a)
	elif b <= 1.6e+27:
		tmp = 0.5 / (a / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.05e+112)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 1.6e+27)
		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(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.05e+112)
		tmp = 0.0 - (b / a);
	elseif (b <= 1.6e+27)
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.05e+112], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+27], 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[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0499999999999998e112

   1. Initial program 69.3%

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

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

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

   3. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 96.9%

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

   7. Simplified 96.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 96.9%

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

   9. Applied egg-rr 96.9%

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

   if -3.0499999999999998e112 < b < 1.60000000000000008e27

   1. Initial program 72.3%

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

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

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

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 72.2%

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

   6. Applied egg-rr 72.2%

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

   if 1.60000000000000008e27 < b

   1. Initial program 12.6%

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

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

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

   3. Simplified 12.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 12.6%

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

   6. Applied egg-rr 12.6%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 88.1%

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

   9. Simplified 88.1%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 89.1%

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

   11. Applied egg-rr 89.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{+112}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+75}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+75)
   (- 0.0 (/ b a))
   (if (<= b 6.5e+31)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+75) {
		tmp = 0.0 - (b / a);
	} else if (b <= 6.5e+31) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-8d+75)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 6.5d+31) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+75) {
		tmp = 0.0 - (b / a);
	} else if (b <= 6.5e+31) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e+75:
		tmp = 0.0 - (b / a)
	elif b <= 6.5e+31:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+75)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 6.5e+31)
		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(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e+75)
		tmp = 0.0 - (b / a);
	elseif (b <= 6.5e+31)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+75], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+31], 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[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.99999999999999941e75

   1. Initial program 72.1%

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

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

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 97.2%

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

   7. Simplified 97.2%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 97.2%

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

   9. Applied egg-rr 97.2%

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

   if -7.99999999999999941e75 < b < 6.5000000000000004e31

   1. Initial program 70.7%

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

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

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

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 70.6%

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

   6. Applied egg-rr 70.6%

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

   if 6.5000000000000004e31 < b

   1. Initial program 10.4%

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

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

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

   3. Simplified 10.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 10.3%

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

   6. Applied egg-rr 10.3%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 89.2%

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

   9. Simplified 89.2%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 90.2%

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

   11. Applied egg-rr 90.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+75}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{-37}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.9e-37)
   (- 0.0 (/ b a))
   (if (<= b 1.6e+27)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e-37) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-5.9d-37)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 1.6d+27) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e-37) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.9e-37:
		tmp = 0.0 - (b / a)
	elif b <= 1.6e+27:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.9e-37)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 1.6e+27)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.9e-37)
		tmp = 0.0 - (b / a);
	elseif (b <= 1.6e+27)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.9e-37], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+27], 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[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8999999999999996e-37

   1. Initial program 74.9%

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

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

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

   3. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 89.0%

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

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 89.0%

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

   9. Applied egg-rr 89.0%

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

   if -5.8999999999999996e-37 < b < 1.60000000000000008e27

   1. Initial program 68.3%

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

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

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

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 66.3%

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

   7. Simplified 66.3%

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

   if 1.60000000000000008e27 < b

   1. Initial program 12.6%

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

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

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

   3. Simplified 12.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 12.6%

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

   6. Applied egg-rr 12.6%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 88.1%

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

   9. Simplified 88.1%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 89.1%

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

   11. Applied egg-rr 89.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{-37}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-36}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-36)
   (- 0.0 (/ b a))
   (if (<= b 1.6e+27)
     (/ 0.5 (/ a (- (sqrt (* -4.0 (* a c))) b)))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-36) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = 0.5 / (a / (sqrt((-4.0 * (a * c))) - b));
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-8d-36)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 1.6d+27) then
        tmp = 0.5d0 / (a / (sqrt(((-4.0d0) * (a * c))) - b))
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-36) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = 0.5 / (a / (Math.sqrt((-4.0 * (a * c))) - b));
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-36:
		tmp = 0.0 - (b / a)
	elif b <= 1.6e+27:
		tmp = 0.5 / (a / (math.sqrt((-4.0 * (a * c))) - b))
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-36)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 1.6e+27)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b)));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-36)
		tmp = 0.0 - (b / a);
	elseif (b <= 1.6e+27)
		tmp = 0.5 / (a / (sqrt((-4.0 * (a * c))) - b));
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-36], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+27], N[(0.5 / N[(a / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.9999999999999995e-36

   1. Initial program 74.9%

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

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

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

   3. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 89.0%

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

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 89.0%

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

   9. Applied egg-rr 89.0%

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

   if -7.9999999999999995e-36 < b < 1.60000000000000008e27

   1. Initial program 68.3%

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

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

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

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 68.3%

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

   6. Applied egg-rr 68.3%

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

   7. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.3%

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

   9. Simplified 66.3%

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

   if 1.60000000000000008e27 < b

   1. Initial program 12.6%

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

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

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

   3. Simplified 12.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 12.6%

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

   6. Applied egg-rr 12.6%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 88.1%

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

   9. Simplified 88.1%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 89.1%

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

   11. Applied egg-rr 89.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-36}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.4 \cdot 10^{-32}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.4e-32)
   (- 0.0 (/ b a))
   (if (<= b 1.6e+27)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.4e-32) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-9.4d-32)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 1.6d+27) then
        tmp = (0.5d0 / a) * (sqrt((a * (c * (-4.0d0)))) - b)
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.4e-32) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.6e+27) {
		tmp = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.4e-32:
		tmp = 0.0 - (b / a)
	elif b <= 1.6e+27:
		tmp = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.4e-32)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 1.6e+27)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.4e-32)
		tmp = 0.0 - (b / a);
	elseif (b <= 1.6e+27)
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.4e-32], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+27], 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[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.40000000000000039e-32

   1. Initial program 74.9%

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

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

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

   3. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 89.0%

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

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 89.0%

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

   9. Applied egg-rr 89.0%

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

   if -9.40000000000000039e-32 < b < 1.60000000000000008e27

   1. Initial program 68.3%

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

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

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

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 68.2%

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

   6. Applied egg-rr 68.2%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 66.2%

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

   9. Simplified 66.2%

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

   if 1.60000000000000008e27 < b

   1. Initial program 12.6%

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

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

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

   3. Simplified 12.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 12.6%

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

   6. Applied egg-rr 12.6%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 88.1%

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

   9. Simplified 88.1%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 89.1%

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

   11. Applied egg-rr 89.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.4 \cdot 10^{-32}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-297)
   (- 0.0 (/ b a))
   (/ (/ 0.5 (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-297) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / 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 <= (-1.9d-297)) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = (0.5d0 / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-297) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-297:
		tmp = 0.0 - (b / a)
	else:
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-297)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-297)
		tmp = 0.0 - (b / a);
	else
		tmp = (0.5 / (((0.5 / (b / a)) / b) + (-0.5 / c))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-297], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.90000000000000002e-297

   1. Initial program 76.5%

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

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

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

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 63.9%

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

   7. Simplified 63.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 63.9%

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

   9. Applied egg-rr 63.9%

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

   if -1.90000000000000002e-297 < b

   1. Initial program 35.2%

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

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

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

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 35.2%

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

   6. Applied egg-rr 35.2%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 55.5%

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

   9. Simplified 55.5%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 58.2%

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

   11. Applied egg-rr 58.2%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-297)
   (- 0.0 (/ b a))
   (/ (/ 0.5 b) (+ (/ (/ 0.5 (/ b a)) b) (/ -0.5 c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-297) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = (0.5 / b) / (((0.5 / (b / a)) / b) + (-0.5 / c));
	}
	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 <= (-1.9d-297)) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = (0.5d0 / b) / (((0.5d0 / (b / a)) / b) + ((-0.5d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-297) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = (0.5 / b) / (((0.5 / (b / a)) / b) + (-0.5 / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-297:
		tmp = 0.0 - (b / a)
	else:
		tmp = (0.5 / b) / (((0.5 / (b / a)) / b) + (-0.5 / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-297)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(Float64(0.5 / Float64(b / a)) / b) + Float64(-0.5 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-297)
		tmp = 0.0 - (b / a);
	else
		tmp = (0.5 / b) / (((0.5 / (b / a)) / b) + (-0.5 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-297], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(N[(0.5 / N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.90000000000000002e-297

   1. Initial program 76.5%

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

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

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

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 63.9%

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

   7. Simplified 63.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 63.9%

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

   9. Applied egg-rr 63.9%

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

   if -1.90000000000000002e-297 < b

   1. Initial program 35.2%

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

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

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

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 35.2%

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

   6. Applied egg-rr 35.2%

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

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 55.5%

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

   9. Simplified 55.5%

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

       1. associate-/r* N/A

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

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

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

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

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

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

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

       5. associate-/r* N/A

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

       6. associate-*r/ N/A

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

       7. clear-num N/A

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

       8. div-inv N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 58.1%

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

   11. Applied egg-rr 58.1%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{b}}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{\frac{0.5}{\frac{b}{a}}}{b} + \frac{-0.5}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- 0.0 (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0 - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(0.0 - Float64(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 <= -5e-310)
		tmp = 0.0 - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 76.9%

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

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

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

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 62.9%

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

   7. Simplified 62.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 62.9%

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

   9. Applied egg-rr 62.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

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

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

   3. Simplified 34.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 58.7%

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

   7. Simplified 58.7%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- 0.0 (/ b a)) (* c (/ -1.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-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 <= (-5d-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 <= -5e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c * (-1.0 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-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 <= -5e-310)
		tmp = Float64(0.0 - Float64(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 <= -5e-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, -5e-310], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 76.9%

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

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

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

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 62.9%

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

   7. Simplified 62.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 62.9%

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

   9. Applied egg-rr 62.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

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

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

   3. Simplified 34.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{b}\right)\right)}\right)\right) \]

   7. Simplified 42.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-2 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{a}{b \cdot \left(b \cdot b\right)}\right) + \frac{-1}{b}\right)} \]

   8. Taylor expanded in c around 0

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-1}{b}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 58.6%

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

   10. Simplified 58.6%

       \[\leadsto c \cdot \color{blue}{\frac{-1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.58 \cdot 10^{+66}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.58e+66) (- 0.0 (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.58e+66) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 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 <= 1.58d+66) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.58e+66) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.58e+66:
		tmp = 0.0 - (b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.58e+66)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.58e+66)
		tmp = 0.0 - (b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.58e+66], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.58e66

   1. Initial program 68.3%

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

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

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

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   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 40.8%

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

   7. Simplified 40.8%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   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 40.8%

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

   9. Applied egg-rr 40.8%

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

   if 1.58e66 < b

   1. Initial program 9.9%

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

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

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

   3. Simplified 9.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   6. Step-by-step derivation

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\left(a \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot b\right)\right)\right), b\right) \]

      13. *-lowering-*.f64 72.6%

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

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{\left(0 - c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}} \]
   8. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(0 - c\right) + \left(\mathsf{neg}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)\right), b\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) + \left(0 - c\right)\right), b\right) \]

   9. Applied egg-rr 30.5%

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

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{c}, b\right) \]
   11. Step-by-step derivation

   12. Simplified 30.4%

       \[\leadsto \frac{\color{blue}{c}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.58 \cdot 10^{+66}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.1% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

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

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

3. Simplified 55.0%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \left(\frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\left(a \cdot {c}^{2}\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right), \left(b \cdot b\right)\right)\right), b\right) \]

   13. *-lowering-*.f64 23.7%

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

7. Simplified 23.7%

   \[\leadsto \color{blue}{\frac{\left(0 - c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}} \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(0 - c\right) + \left(\mathsf{neg}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)\right), b\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) + \left(0 - c\right)\right), b\right) \]

9. Applied egg-rr 8.7%

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

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{c}, b\right) \]
11. Step-by-step derivation

12. Simplified 9.1%

    \[\leadsto \frac{\color{blue}{c}}{b} \]
13. Add Preprocessing

Alternative 13: 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 55.0%

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

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

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

3. Simplified 55.0%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 55.0%

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

6. Applied egg-rr 55.0%

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

7. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. distribute-lft-neg-in N/A

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

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

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

   4. associate-*r/ N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. distribute-lft-neg-in N/A

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

   9. metadata-eval N/A

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

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

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

   11. /-lowering-/.f64 31.3%

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

9. Simplified 31.3%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{b \cdot -0.5}{c} + 0.5 \cdot \frac{a}{b}}} \]

10. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 2.5%

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

12. Simplified 2.5%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
13. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024288 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))