quadp (p42, positive)

Percentage Accurate: 51.4% → 85.8%

Time: 12.5s

Alternatives: 10

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: 51.4% 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: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \left|b \cdot \sqrt{1 + \frac{\frac{c}{\frac{b}{a}}}{\frac{b}{-4}}}\right|\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+118)
   (*
    (/ -0.5 a)
    (- b (fabs (* b (sqrt (+ 1.0 (/ (/ c (/ b a)) (/ b -4.0))))))))
   (if (<= b 2.6e-73)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = (-0.5 / a) * (b - fabs((b * sqrt((1.0 + ((c / (b / a)) / (b / -4.0)))))));
	} else if (b <= 2.6e-73) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} 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+118)) then
        tmp = ((-0.5d0) / a) * (b - abs((b * sqrt((1.0d0 + ((c / (b / a)) / (b / (-4.0d0))))))))
    else if (b <= 2.6d-73) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    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+118) {
		tmp = (-0.5 / a) * (b - Math.abs((b * Math.sqrt((1.0 + ((c / (b / a)) / (b / -4.0)))))));
	} else if (b <= 2.6e-73) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+118:
		tmp = (-0.5 / a) * (b - math.fabs((b * math.sqrt((1.0 + ((c / (b / a)) / (b / -4.0)))))))
	elif b <= 2.6e-73:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+118)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - abs(Float64(b * sqrt(Float64(1.0 + Float64(Float64(c / Float64(b / a)) / Float64(b / -4.0))))))));
	elseif (b <= 2.6e-73)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+118)
		tmp = (-0.5 / a) * (b - abs((b * sqrt((1.0 + ((c / (b / a)) / (b / -4.0)))))));
	elseif (b <= 2.6e-73)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+118], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Abs[N[(b * N[Sqrt[N[(1.0 + N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] / N[(b / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-73], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999972e118

   1. Initial program 29.0%

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

   4. Applied egg-rr 28.9%

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

   5. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

      13. /-lowering-/.f64 29.1%

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

   7. Simplified 29.1%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(b \cdot b\right) \cdot \left(1 + \left(a \cdot \frac{c}{b}\right) \cdot \frac{-4}{b}\right)}}\right) \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. rem-sqrt-square N/A

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

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

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

      4. sqrt-prod N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

      13. *-commutative N/A

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

      14. div-inv N/A

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

      15. associate-*l* N/A

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

      16. associate-/r/ N/A

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

      17. un-div-inv N/A

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

      18. /-lowering-/.f64 N/A

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

      19. /-lowering-/.f64 N/A

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

      20. /-lowering-/.f64 99.5%

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

   9. Applied egg-rr 99.5%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \color{blue}{\left|b \cdot \sqrt{1 + \frac{\frac{c}{\frac{b}{a}}}{\frac{b}{-4}}}\right|}\right) \]

   if -4.99999999999999972e118 < b < 2.6000000000000001e-73

   1. Initial program 91.5%

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

   if 2.6000000000000001e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \left|b \cdot \sqrt{1 + \frac{\frac{c}{\frac{b}{a}}}{\frac{b}{-4}}}\right|\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+118)
   (- (/ c b) (/ b a))
   (if (<= b 1.35e-78)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.35e-78) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} 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+118)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.35d-78) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    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+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.35e-78) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+118:
		tmp = (c / b) - (b / a)
	elif b <= 1.35e-78:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+118)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.35e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+118)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.35e-78)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+118], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-78], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999972e118

   1. Initial program 29.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 95.9%

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

   8. Simplified 95.9%

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

   if -4.99999999999999972e118 < b < 1.34999999999999997e-78

   1. Initial program 91.5%

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

   if 1.34999999999999997e-78 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e+98)
   (- (/ c b) (/ b a))
   (if (<= b 8.6e-76)
     (* (/ -0.5 a) (- b (sqrt (+ (* b b) (* a (* c -4.0))))))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e+98) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-76) {
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	} 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 <= (-9.2d+98)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8.6d-76) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((b * b) + (a * (c * (-4.0d0))))))
    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 <= -9.2e+98) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-76) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.2e+98:
		tmp = (c / b) - (b / a)
	elif b <= 8.6e-76:
		tmp = (-0.5 / a) * (b - math.sqrt(((b * b) + (a * (c * -4.0)))))
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.2e+98)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.6e-76)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.2e+98)
		tmp = (c / b) - (b / a);
	elseif (b <= 8.6e-76)
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.2e+98], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-76], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.20000000000000053e98

   1. Initial program 38.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 96.4%

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

   8. Simplified 96.4%

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

   if -9.20000000000000053e98 < b < 8.5999999999999998e-76

   1. Initial program 91.0%

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

   4. Applied egg-rr 90.9%

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

   if 8.5999999999999998e-76 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-44)
   (- (/ c b) (/ b a))
   (if (<= b 4.5e-74)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-74) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} 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 <= (-3d-44)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.5d-74) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -3e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-74) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-44:
		tmp = (c / b) - (b / a)
	elif b <= 4.5e-74:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-44)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.5e-74)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-44)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.5e-74)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-44], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-74], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000002e-44

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.7%

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

   8. Simplified 90.7%

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

   if -3.0000000000000002e-44 < b < 4.4999999999999999e-74

   1. Initial program 87.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 84.6%

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

   5. Simplified 84.6%

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

   if 4.4999999999999999e-74 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-44)
   (- (/ c b) (/ b a))
   (if (<= b 8.5e-76)
     (* (/ -0.5 a) (- b (sqrt (* c (* a -4.0)))))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-76) {
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	} 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 <= (-6.5d-44)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8.5d-76) then
        tmp = ((-0.5d0) / a) * (b - sqrt((c * (a * (-4.0d0)))))
    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 <= -6.5e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-76) {
		tmp = (-0.5 / a) * (b - Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-44:
		tmp = (c / b) - (b / a)
	elif b <= 8.5e-76:
		tmp = (-0.5 / a) * (b - math.sqrt((c * (a * -4.0))))
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-44)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.5e-76)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-44)
		tmp = (c / b) - (b / a);
	elseif (b <= 8.5e-76)
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-44], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-76], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-44

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.7%

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

   8. Simplified 90.7%

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

   if -6.5e-44 < b < 8.50000000000000038e-76

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

      13. /-lowering-/.f64 24.6%

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

   7. Simplified 24.6%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{\left(b \cdot b\right) \cdot \left(1 + \left(a \cdot \frac{c}{b}\right) \cdot \frac{-4}{b}\right)}}\right) \]

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 84.5%

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

   10. Simplified 84.5%

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

   if 8.50000000000000038e-76 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-43)
   (- (/ c b) (/ b a))
   (if (<= b 7.2e-76)
     (* (/ -0.5 a) (- b (sqrt (* -4.0 (* a c)))))
     (/ (- 0.0 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-43) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.2e-76) {
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	} 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 <= (-1.15d-43)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.2d-76) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((-4.0d0) * (a * c))))
    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 <= -1.15e-43) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.2e-76) {
		tmp = (-0.5 / a) * (b - Math.sqrt((-4.0 * (a * c))));
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-43:
		tmp = (c / b) - (b / a)
	elif b <= 7.2e-76:
		tmp = (-0.5 / a) * (b - math.sqrt((-4.0 * (a * c))))
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-43)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.2e-76)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-43)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.2e-76)
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-43], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-76], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1499999999999999e-43

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.7%

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

   8. Simplified 90.7%

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

   if -1.1499999999999999e-43 < b < 7.2000000000000001e-76

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 84.5%

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

   7. Simplified 84.5%

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

   if 7.2000000000000001e-76 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-299) (/ (- 0.0 b) a) (/ (- 0.0 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-299) {
		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 <= 2d-299) 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 <= 2e-299) {
		tmp = (0.0 - b) / a;
	} else {
		tmp = (0.0 - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-299:
		tmp = (0.0 - b) / a
	else:
		tmp = (0.0 - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-299)
		tmp = Float64(Float64(0.0 - b) / a);
	else
		tmp = Float64(Float64(0.0 - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-299)
		tmp = (0.0 - b) / a;
	else
		tmp = (0.0 - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-299], N[(N[(0.0 - b), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999998e-299

   1. Initial program 71.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. --lowering--.f64 63.3%

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

   5. Simplified 63.3%

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

   if 1.99999999999999998e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 70.5%

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

   5. Simplified 70.5%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{0 - c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- 0.0 c) b))

C

double code(double a, double b, double c) {
	return (0.0 - 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 = (0.0d0 - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return (0.0 - c) / b;
}

Python

def code(a, b, c):
	return (0.0 - c) / b

Julia

function code(a, b, c)
	return Float64(Float64(0.0 - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.0 - c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(0.0 - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 38.3%

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

5. Simplified 38.3%

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

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 38.3%

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

7. Applied egg-rr 38.3%

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

8. Final simplification 38.3%

   \[\leadsto \frac{0 - c}{b} \]
9. Add Preprocessing

Alternative 9: 10.7% 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 51.4%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   6. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 21.7%

   \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

6. Taylor expanded in b around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. metadata-eval N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 29.2%

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

8. Simplified 29.2%

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

9. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 13.4%

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

11. Simplified 13.4%

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

Alternative 10: 2.6% 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 51.4%

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

4. Applied egg-rr 35.0%

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

5. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 2.7%

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

7. Simplified 2.7%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. 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 2024192 
(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)))