Quadratic roots, full range

Percentage Accurate: 51.7% → 84.1%

Time: 36.0s

Alternatives: 8

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: 51.7% 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: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ t_1 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)\right|}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+35}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b)))
        (t_1 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= b -1.5e+87)
     (/ (fabs (+ b (fma -2.0 (* a (/ c b)) b))) (* a 2.0))
     (if (<= b 4e-127)
       t_1
       (if (<= b 1e-58)
         t_0
         (if (<= b 1.65e-13)
           t_1
           (if (<= b 1.55e+35)
             (*
              c
              (+
               (*
                c
                (-
                 (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0)))
                 (/ a (pow b 3.0))))
               (/ -1.0 b)))
             t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.5e+87) {
		tmp = fabs((b + fma(-2.0, (a * (c / b)), b))) / (a * 2.0);
	} else if (b <= 4e-127) {
		tmp = t_1;
	} else if (b <= 1e-58) {
		tmp = t_0;
	} else if (b <= 1.65e-13) {
		tmp = t_1;
	} else if (b <= 1.55e+35) {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(abs(Float64(b + fma(-2.0, Float64(a * Float64(c / b)), b))) / Float64(a * 2.0));
	elseif (b <= 4e-127)
		tmp = t_1;
	elseif (b <= 1e-58)
		tmp = t_0;
	elseif (b <= 1.65e-13)
		tmp = t_1;
	elseif (b <= 1.55e+35)
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e+87], N[(N[Abs[N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-127], t$95$1, If[LessEqual[b, 1e-58], t$95$0, If[LessEqual[b, 1.65e-13], t$95$1, If[LessEqual[b, 1.55e+35], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.4999999999999999e87

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 2.2%

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

      3. pow2 2.2%

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

      4. add-sqr-sqrt 18.1%

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

      5. sqrt-unprod 29.9%

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

      6. sqr-neg 29.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 49.1%

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

      9. +-commutative 49.1%

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

      10. fma-define 49.1%

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

      11. associate-/l* 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. rem-sqrt-square 92.0%

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

   9. Simplified 92.0%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127 or 1e-58 < b < 1.65e-13

   1. Initial program 83.8%

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

   if 4.0000000000000001e-127 < b < 1e-58 or 1.54999999999999993e35 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. associate-*r/ 83.4%

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

      2. mul-1-neg 83.4%

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

   7. Simplified 83.4%

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

   if 1.65e-13 < b < 1.54999999999999993e35

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

   3. Simplified 19.5%

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

   5. Taylor expanded in c around 0 92.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)\right|}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+35}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)\right|}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+87)
   (/ (fabs (+ b (fma -2.0 (* a (/ c b)) b))) (* a 2.0))
   (if (<= b 4e-127)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+87) {
		tmp = fabs((b + fma(-2.0, (a * (c / b)), b))) / (a * 2.0);
	} else if (b <= 4e-127) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(abs(Float64(b + fma(-2.0, Float64(a * Float64(c / b)), b))) / Float64(a * 2.0));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+87], N[(N[Abs[N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $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 < -1.4999999999999999e87

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 2.2%

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

      3. pow2 2.2%

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

      4. add-sqr-sqrt 18.1%

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

      5. sqrt-unprod 29.9%

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

      6. sqr-neg 29.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 49.1%

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

      9. +-commutative 49.1%

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

      10. fma-define 49.1%

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

      11. associate-/l* 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. rem-sqrt-square 92.0%

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

   9. Simplified 92.0%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127

   1. Initial program 84.5%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left|b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)\right|}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{0.5 \cdot \left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+87)
   (/ (* 0.5 (fabs (* -2.0 (- (* a (/ c b)) b)))) a)
   (if (<= b 4e-127)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+87) {
		tmp = (0.5 * fabs((-2.0 * ((a * (c / b)) - b)))) / a;
	} else if (b <= 4e-127) {
		tmp = (sqrt(((b * b) - (c * (a * 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 <= (-1.5d+87)) then
        tmp = (0.5d0 * abs(((-2.0d0) * ((a * (c / b)) - b)))) / a
    else if (b <= 4d-127) then
        tmp = (sqrt(((b * b) - (c * (a * 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 <= -1.5e+87) {
		tmp = (0.5 * Math.abs((-2.0 * ((a * (c / b)) - b)))) / a;
	} else if (b <= 4e-127) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e+87:
		tmp = (0.5 * math.fabs((-2.0 * ((a * (c / b)) - b)))) / a
	elif b <= 4e-127:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(Float64(0.5 * abs(Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)))) / a);
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+87)
		tmp = (0.5 * abs((-2.0 * ((a * (c / b)) - b)))) / a;
	elseif (b <= 4e-127)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+87], N[(N[(0.5 * N[Abs[N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $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 < -1.4999999999999999e87

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 2.2%

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

      3. pow2 2.2%

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

      4. add-sqr-sqrt 18.1%

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

      5. sqrt-unprod 29.9%

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

      6. sqr-neg 29.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 49.1%

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

      9. +-commutative 49.1%

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

      10. fma-define 49.1%

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

      11. associate-/l* 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. rem-sqrt-square 92.0%

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

   9. Simplified 92.0%

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

   10. Taylor expanded in b around 0 83.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]
   11. Step-by-step derivation

       1. associate-*r/ 84.1%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]

       2. metadata-eval 84.1%

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

       3. cancel-sign-sub-inv 84.1%

          \[\leadsto \frac{0.5 \cdot \left|\color{blue}{-2 \cdot \frac{a \cdot c}{b} - -2 \cdot b}\right|}{a} \]

       4. distribute-lft-out-- 84.1%

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

       5. associate-*r/ 91.9%

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

   12. Simplified 91.9%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127

   1. Initial program 84.5%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{0.5 \cdot \left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-5)
   (/ b (- a))
   (if (<= b 4e-127) (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-5) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		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 <= (-2.9d-5)) then
        tmp = b / -a
    else if (b <= 4d-127) 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 <= -2.9e-5) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		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 <= -2.9e-5:
		tmp = b / -a
	elif b <= 4e-127:
		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 <= -2.9e-5)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-5)
		tmp = b / -a;
	elseif (b <= 4e-127)
		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, -2.9e-5], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-127], 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 < -2.9e-5

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in b around -inf 87.8%

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

      1. associate-*r/ 87.8%

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

      2. mul-1-neg 87.8%

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

   7. Simplified 87.8%

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

   if -2.9e-5 < b < 4.0000000000000001e-127

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -215000:\\ \;\;\;\;\frac{0.5 \cdot \left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -215000.0)
   (/ (* 0.5 (fabs (* -2.0 (- (* a (/ c b)) b)))) a)
   (if (<= b 4e-127) (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -215000.0) {
		tmp = (0.5 * fabs((-2.0 * ((a * (c / b)) - b)))) / a;
	} else if (b <= 4e-127) {
		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 <= (-215000.0d0)) then
        tmp = (0.5d0 * abs(((-2.0d0) * ((a * (c / b)) - b)))) / a
    else if (b <= 4d-127) 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 <= -215000.0) {
		tmp = (0.5 * Math.abs((-2.0 * ((a * (c / b)) - b)))) / a;
	} else if (b <= 4e-127) {
		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 <= -215000.0:
		tmp = (0.5 * math.fabs((-2.0 * ((a * (c / b)) - b)))) / a
	elif b <= 4e-127:
		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 <= -215000.0)
		tmp = Float64(Float64(0.5 * abs(Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)))) / a);
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -215000.0)
		tmp = (0.5 * abs((-2.0 * ((a * (c / b)) - b)))) / a;
	elseif (b <= 4e-127)
		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, -215000.0], N[(N[(0.5 * N[Abs[N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4e-127], 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 < -215000

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in a around 0 1.9%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 2.3%

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

      3. pow2 2.3%

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

      4. add-sqr-sqrt 15.4%

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

      5. sqrt-unprod 24.5%

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

      6. sqr-neg 24.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 54.9%

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

      9. +-commutative 54.9%

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

      10. fma-define 54.9%

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

      11. associate-/l* 54.9%

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

   7. Applied egg-rr 54.9%

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

      1. unpow2 54.9%

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

      2. rem-sqrt-square 89.1%

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

   9. Simplified 89.1%

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

   10. Taylor expanded in b around 0 82.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]
   11. Step-by-step derivation

       1. associate-*r/ 82.9%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]

       2. metadata-eval 82.9%

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

       3. cancel-sign-sub-inv 82.9%

          \[\leadsto \frac{0.5 \cdot \left|\color{blue}{-2 \cdot \frac{a \cdot c}{b} - -2 \cdot b}\right|}{a} \]

       4. distribute-lft-out-- 82.9%

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

       5. associate-*r/ 89.1%

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

   12. Simplified 89.1%

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

   if -215000 < b < 4.0000000000000001e-127

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in b around 0 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*r* 75.0%

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

   7. Simplified 75.0%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -215000:\\ \;\;\;\;\frac{0.5 \cdot \left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-295) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-295) {
		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 <= 2.1d-295) 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 <= 2.1e-295) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-295:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-295)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-295)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-295], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999993e-295

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. mul-1-neg 66.7%

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

   7. Simplified 66.7%

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

   if 2.09999999999999993e-295 < b

   1. Initial program 34.9%

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

      1. *-commutative 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. mul-1-neg 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 29.0× 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 / Float64(-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 50.5%

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

   1. *-commutative 50.5%

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

3. Simplified 50.5%

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

5. Taylor expanded in b around -inf 33.6%

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

   1. associate-*r/ 33.6%

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

   2. mul-1-neg 33.6%

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

7. Simplified 33.6%

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

8. Final simplification 33.6%

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

Alternative 8: 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 50.5%

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

   1. *-commutative 50.5%

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

3. Simplified 50.5%

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

   1. sub-neg 50.5%

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

   2. fma-undefine 50.5%

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

   3. add-sqr-sqrt 41.7%

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

   4. hypot-define 52.4%

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

   5. *-commutative 52.4%

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

   6. associate-*r* 52.4%

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

   7. *-commutative 52.4%

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

   8. add-sqr-sqrt 29.8%

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

   9. sqrt-unprod 40.1%

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

   10. sqr-neg 40.1%

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

   11. sqrt-prod 15.8%

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

   12. add-sqr-sqrt 27.9%

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

6. Applied egg-rr 27.9%

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

7. Taylor expanded in b around inf 2.7%

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

8. Final simplification 2.7%

   \[\leadsto \frac{b}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))