Quadratic roots, full range

Percentage Accurate: 51.2% → 86.0%

Time: 13.5s

Alternatives: 9

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot {b}^{-2} + \frac{-1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+138)
   (/ b (- a))
   (if (<= b 1.9e-92)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ (/ 1.0 b) (+ (* a (pow b -2.0)) (/ -1.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+138) {
		tmp = b / -a;
	} else if (b <= 1.9e-92) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a * pow(b, -2.0)) + (-1.0 / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+138)) then
        tmp = b / -a
    else if (b <= 1.9d-92) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / b) / ((a * (b ** (-2.0d0))) + ((-1.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+138) {
		tmp = b / -a;
	} else if (b <= 1.9e-92) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a * Math.pow(b, -2.0)) + (-1.0 / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+138:
		tmp = b / -a
	elif b <= 1.9e-92:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / b) / ((a * math.pow(b, -2.0)) + (-1.0 / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+138)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.9e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / b) / Float64(Float64(a * (b ^ -2.0)) + Float64(-1.0 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+138)
		tmp = b / -a;
	elseif (b <= 1.9e-92)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = (1.0 / b) / ((a * (b ^ -2.0)) + (-1.0 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+138], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.9e-92], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] / N[(N[(a * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e138

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around -inf 90.8%

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

      1. associate-*r/ 90.8%

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

      2. mul-1-neg 90.8%

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

   7. Simplified 90.8%

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

   if -2.0000000000000001e138 < b < 1.9e-92

   1. Initial program 88.8%

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

   if 1.9e-92 < b

   1. Initial program 18.8%

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

      1. *-commutative 18.8%

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

   3. Simplified 18.8%

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

   6. Applied egg-rr 10.2%

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

      1. unpow-1 10.2%

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

      2. associate-/l* 10.2%

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

   8. Simplified 10.2%

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

   9. Taylor expanded in b around -inf 19.9%

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

       1. associate-*r* 19.9%

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

       2. neg-mul-1 19.9%

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

   11. Simplified 19.9%

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

       1. *-un-lft-identity 19.9%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 59.5%

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

       4. sqr-neg 59.5%

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

       5. sqrt-prod 83.9%

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

       6. add-sqr-sqrt 84.2%

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

       7. div-inv 84.2%

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

       8. pow-flip 84.2%

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

       9. metadata-eval 84.2%

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

       10. inv-pow 84.2%

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

   13. Applied egg-rr 84.2%

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

       1. *-lft-identity 84.2%

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

       2. associate-/r* 86.3%

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

       3. unpow-1 86.3%

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

   15. Simplified 86.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot {b}^{-2} + \frac{-1}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot {b}^{-2} + \frac{-1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-52)
   (/ b (- a))
   (if (<= b 7.9e-94)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (/ 1.0 b) (+ (* a (pow b -2.0)) (/ -1.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-52) {
		tmp = b / -a;
	} else if (b <= 7.9e-94) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a * pow(b, -2.0)) + (-1.0 / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.2d-52)) then
        tmp = b / -a
    else if (b <= 7.9d-94) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / b) / ((a * (b ** (-2.0d0))) + ((-1.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-52) {
		tmp = b / -a;
	} else if (b <= 7.9e-94) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a * Math.pow(b, -2.0)) + (-1.0 / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-52:
		tmp = b / -a
	elif b <= 7.9e-94:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / b) / ((a * math.pow(b, -2.0)) + (-1.0 / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-52)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 7.9e-94)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / b) / Float64(Float64(a * (b ^ -2.0)) + Float64(-1.0 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.2e-52)
		tmp = b / -a;
	elseif (b <= 7.9e-94)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = (1.0 / b) / ((a * (b ^ -2.0)) + (-1.0 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-52], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 7.9e-94], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] / N[(N[(a * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2000000000000001e-52

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in b around -inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. mul-1-neg 85.4%

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

   7. Simplified 85.4%

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

   if -3.2000000000000001e-52 < b < 7.9e-94

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in a around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-*r* 79.1%

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

   7. Simplified 79.1%

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

   if 7.9e-94 < b

   1. Initial program 18.8%

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

      1. *-commutative 18.8%

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

   3. Simplified 18.8%

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

   6. Applied egg-rr 10.2%

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

      1. unpow-1 10.2%

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

      2. associate-/l* 10.2%

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

   8. Simplified 10.2%

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

   9. Taylor expanded in b around -inf 19.9%

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

       1. associate-*r* 19.9%

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

       2. neg-mul-1 19.9%

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

   11. Simplified 19.9%

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

       1. *-un-lft-identity 19.9%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 59.5%

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

       4. sqr-neg 59.5%

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

       5. sqrt-prod 83.9%

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

       6. add-sqr-sqrt 84.2%

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

       7. div-inv 84.2%

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

       8. pow-flip 84.2%

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

       9. metadata-eval 84.2%

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

       10. inv-pow 84.2%

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

   13. Applied egg-rr 84.2%

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

       1. *-lft-identity 84.2%

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

       2. associate-/r* 86.3%

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

       3. unpow-1 86.3%

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

   15. Simplified 86.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot {b}^{-2} + \frac{-1}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.3e-62)
   (/ b (- a))
   (if (<= b 2.1e-92)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.3e-62) {
		tmp = b / -a;
	} else if (b <= 2.1e-92) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.3d-62)) then
        tmp = b / -a
    else if (b <= 2.1d-92) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.3e-62) {
		tmp = b / -a;
	} else if (b <= 2.1e-92) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.3e-62:
		tmp = b / -a
	elif b <= 2.1e-92:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.3e-62)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.1e-92)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.3e-62)
		tmp = b / -a;
	elseif (b <= 2.1e-92)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.3e-62], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.1e-92], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2999999999999997e-62

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in b around -inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. mul-1-neg 85.4%

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

   7. Simplified 85.4%

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

   if -5.2999999999999997e-62 < b < 2.1e-92

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in a around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-*r* 79.1%

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

   7. Simplified 79.1%

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

   if 2.1e-92 < b

   1. Initial program 18.8%

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

      1. *-commutative 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in a around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. mul-1-neg 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-120)
   (/ b (- a))
   (if (<= b 6.2e-93)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-120) {
		tmp = b / -a;
	} else if (b <= 6.2e-93) {
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-120)) then
        tmp = b / -a
    else if (b <= 6.2d-93) then
        tmp = (b + sqrt((a * (c * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-120) {
		tmp = b / -a;
	} else if (b <= 6.2e-93) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-120:
		tmp = b / -a
	elif b <= 6.2e-93:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-120)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.2e-93)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-120)
		tmp = b / -a;
	elseif (b <= 6.2e-93)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-120], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.2e-93], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3999999999999999e-120

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b around -inf 81.4%

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

      1. associate-*r/ 81.4%

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

      2. mul-1-neg 81.4%

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

   7. Simplified 81.4%

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

   if -2.3999999999999999e-120 < b < 6.19999999999999999e-93

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in a around inf 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r* 82.8%

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

   7. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. sqrt-prod 48.3%

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

   9. Applied egg-rr 48.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c \cdot -4} \cdot \sqrt{a}} - b}{a \cdot 2} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 48.3%

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

       2. *-commutative 48.3%

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

       3. times-frac 48.3%

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\sqrt{c \cdot -4} \cdot \sqrt{a} - b}{a}} \]

       4. metadata-eval 48.3%

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

       5. sub-neg 48.3%

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

       6. *-commutative 48.3%

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

       7. sqrt-unprod 82.8%

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

       8. add-sqr-sqrt 27.6%

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

       9. sqrt-unprod 82.0%

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

       10. sqr-neg 82.0%

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

       11. sqrt-prod 55.0%

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

       12. add-sqr-sqrt 81.6%

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

   11. Applied egg-rr 81.6%

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

       1. *-commutative 81.6%

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

       2. metadata-eval 81.6%

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

       3. times-frac 81.6%

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

       4. *-rgt-identity 81.6%

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

       5. +-commutative 81.6%

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

   13. Simplified 81.6%

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

   if 6.19999999999999999e-93 < b

   1. Initial program 18.8%

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

      1. *-commutative 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in a around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. mul-1-neg 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\left|b - a \cdot \frac{c}{b}\right|}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ (fabs (- b (* a (/ c b)))) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = fabs((b - (a * (c / b)))) / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = abs((b - (a * (c / b)))) / a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = Math.abs((b - (a * (c / b)))) / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = math.fabs((b - (a * (c / b)))) / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(abs(Float64(b - Float64(a * Float64(c / b)))) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = abs((b - (a * (c / b)))) / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[Abs[N[(b - N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   6. Applied egg-rr 36.9%

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

      1. unpow-1 36.9%

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

      2. associate-/l* 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in a around 0 0.9%

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

       1. mul-1-neg 0.9%

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

       2. unsub-neg 0.9%

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

       3. *-commutative 0.9%

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

   11. Simplified 0.9%

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

       1. clear-num 0.9%

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

       2. inv-pow 0.9%

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

       3. *-commutative 0.9%

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

   13. Applied egg-rr 0.9%

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

       1. unpow-1 0.9%

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

       2. *-commutative 0.9%

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

   15. Simplified 0.9%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 44.5%

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

       3. pow2 44.5%

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

       4. associate-/r/ 44.5%

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

   17. Applied egg-rr 44.5%

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

       1. unpow2 44.5%

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

       2. rem-sqrt-square 66.2%

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

       3. associate-*r* 67.8%

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

       4. *-commutative 67.8%

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

       5. associate-*r/ 67.8%

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

       6. *-rgt-identity 67.8%

          \[\leadsto \frac{\left|b - \frac{\color{blue}{c}}{b} \cdot a\right|}{a} \]

   19. Simplified 67.8%

       \[\leadsto \frac{\color{blue}{\left|b - \frac{c}{b} \cdot a\right|}}{a} \]

   if -5.00000000000023e-311 < b

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in a around 0 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\left|b - a \cdot \frac{c}{b}\right|}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ b (- a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = b / -a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = b / -a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = b / -a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = b / -a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = b / -a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(b / (-a)), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around -inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   if -5.00000000000023e-311 < b

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in a around 0 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2e+32) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e+32) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2d+32) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e+32) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e+32:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e+32)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e+32)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e+32], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000011e32

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in b around -inf 42.9%

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

      1. associate-*r/ 42.9%

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

      2. mul-1-neg 42.9%

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

   7. Simplified 42.9%

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

   if 2.00000000000000011e32 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

   6. Applied egg-rr 5.2%

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

      1. unpow-1 5.2%

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

      2. associate-/l* 5.2%

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

   8. Simplified 5.2%

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

   9. Taylor expanded in b around -inf 25.3%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.8% 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 50.8%

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

   1. *-commutative 50.8%

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

3. Simplified 50.8%

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

6. Applied egg-rr 33.5%

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

   1. unpow-1 33.5%

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

   2. associate-/l* 33.5%

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

8. Simplified 33.5%

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

9. Taylor expanded in b around -inf 10.2%

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

Alternative 9: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

   1. *-commutative 50.8%

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

3. Simplified 50.8%

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

6. Applied egg-rr 33.5%

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

   1. unpow-1 33.5%

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

   2. associate-/l* 33.5%

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

8. Simplified 33.5%

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

9. Taylor expanded in a around 0 2.8%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))