Quadratic roots, narrow range

Percentage Accurate: 55.7% → 92.0%

Time: 14.4s

Alternatives: 12

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.7% 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: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.68:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 0.68)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (-
      (*
       a
       (-
        (*
         a
         (+
          (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
          (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
        (/ (pow c 2.0) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 0.68) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (4.0d0 * (a * c))
    if (b <= 0.68d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 0.68) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (4.0 * (a * c))
	tmp = 0
	if b <= 0.68:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(4.0 * Float64(a * c)))
	tmp = 0.0
	if (b <= 0.68)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (4.0 * (a * c));
	tmp = 0.0;
	if (b <= 0.68)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-5.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.68], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.680000000000000049

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. add-cbrt-cube 85.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 83.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 83.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 83.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 86.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 86.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 86.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 86.1%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 86.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 83.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 88.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 88.1%

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

      11. associate-*l* 88.1%

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

   10. Applied egg-rr 88.1%

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

   if 0.680000000000000049 < b

   1. Initial program 50.6%

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

      1. *-commutative 50.6%

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

      2. +-commutative 50.6%

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

      3. sqr-neg 50.6%

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

      4. unsub-neg 50.6%

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

      5. sqr-neg 50.6%

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

      6. fma-neg 50.7%

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

      7. distribute-lft-neg-in 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

      10. distribute-rgt-neg-in 50.7%

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

      11. metadata-eval 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around 0 95.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around 0 95.7%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 95.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

      2. neg-mul-1 95.7%

         \[\leadsto \frac{\color{blue}{-c}}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   8. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 95.7%

         \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\frac{-1 \cdot {c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   10. Applied egg-rr 95.7%

       \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\frac{-1 \cdot {c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 95.7%

          \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{-1 \cdot \frac{{c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

       2. mul-1-neg 95.7%

          \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

       3. distribute-neg-frac2 95.7%

          \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\frac{{c}^{2}}{-{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   12. Simplified 95.7%

       \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\frac{{c}^{2}}{-{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.68:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 0.6)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (*
      c
      (+
       (*
        c
        (-
         (*
          c
          (+
           (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))
           (* -2.0 (/ (pow a 2.0) (pow b 5.0)))))
         (/ a (pow b 3.0))))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 0.6) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = c * ((c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))) + (-2.0 * (pow(a, 2.0) / pow(b, 5.0))))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (4.0d0 * (a * c))
    if (b <= 0.6d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = c * ((c * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (b ** 5.0d0))))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 0.6) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = c * ((c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (4.0 * (a * c))
	tmp = 0
	if b <= 0.6:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = c * ((c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 7.0))) + (-2.0 * (math.pow(a, 2.0) / math.pow(b, 5.0))))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(4.0 * Float64(a * c)))
	tmp = 0.0
	if (b <= 0.6)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))) + Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 5.0))))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (4.0 * (a * c));
	tmp = 0.0;
	if (b <= 0.6)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = c * ((c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 7.0))) + (-2.0 * ((a ^ 2.0) / (b ^ 5.0))))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.6], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.599999999999999978

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. add-cbrt-cube 85.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 83.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 83.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 83.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 86.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 86.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 86.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 86.1%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 86.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 83.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 88.2%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 88.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 88.1%

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

      11. associate-*l* 88.1%

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

   10. Applied egg-rr 88.1%

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

   if 0.599999999999999978 < b

   1. Initial program 50.6%

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

      1. *-commutative 50.6%

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

      2. +-commutative 50.6%

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

      3. sqr-neg 50.6%

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

      4. unsub-neg 50.6%

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

      5. sqr-neg 50.6%

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

      6. fma-neg 50.7%

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

      7. distribute-lft-neg-in 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

      10. distribute-rgt-neg-in 50.7%

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

      11. metadata-eval 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around 0 95.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around 0 95.5%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{4}}, -{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 4.8)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (*
        a
        (fma -2.0 (/ (* a (pow c 3.0)) (pow b 4.0)) (- (pow (/ c (- b)) 2.0))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 4.0)), -pow((c / -b), 2.0))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(4.0 * Float64(a * c)))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0)), Float64(-(Float64(c / Float64(-b)) ^ 2.0)))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + (-N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. add-cbrt-cube 84.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 84.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 84.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 84.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 84.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 84.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 82.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 87.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 87.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. associate-*l* 87.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. pow1/3 86.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow-pow 86.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. metadata-eval 86.9%

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

      11. associate-*l* 86.9%

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

   10. Applied egg-rr 86.9%

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

   if 4.79999999999999982 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. +-commutative 48.8%

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

      3. sqr-neg 48.8%

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

      4. unsub-neg 48.8%

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

      5. sqr-neg 48.8%

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

      6. fma-neg 48.9%

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

      7. distribute-lft-neg-in 48.9%

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

      8. *-commutative 48.9%

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

      9. *-commutative 48.9%

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

      10. distribute-rgt-neg-in 48.9%

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

      11. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   6. Taylor expanded in a around 0 94.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot c + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
   7. Step-by-step derivation

      1. neg-mul-1 94.1%

         \[\leadsto \frac{\color{blue}{\left(-c\right)} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b} \]

      2. +-commutative 94.1%

         \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \left(-c\right)}}{b} \]

      3. unsub-neg 94.1%

         \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c}}{b} \]

   8. Simplified 94.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{4}}, -{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{4}}, -{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ 1.0 (* 2.0 (/ a (- (sqrt (fma b b (* (* a c) -4.0))) b))))
   (/
    (-
     (*
      a
      (fma -2.0 (/ (* a (pow c 3.0)) (pow b 4.0)) (- (pow (/ c (- b)) 2.0))))
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = 1.0 / (2.0 * (a / (sqrt(fma(b, b, ((a * c) * -4.0))) - b)));
	} else {
		tmp = ((a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 4.0)), -pow((c / -b), 2.0))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(1.0 / Float64(2.0 * Float64(a / Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b))));
	else
		tmp = Float64(Float64(Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0)), Float64(-(Float64(c / Float64(-b)) ^ 2.0)))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(1.0 / N[(2.0 * N[(a / N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + (-N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. add-cbrt-cube 84.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 84.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 84.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 81.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. fma-define 81.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      3. pow2 81.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2}}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      4. pow1/3 80.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      5. pow-pow 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      6. metadata-eval 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      7. associate-*l* 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}{a \cdot 2} \]

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 81.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}} \]

       2. inv-pow 81.2%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]

   12. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

       2. associate-/l* 85.6%

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

       3. fma-undefine 85.6%

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

       4. neg-mul-1 85.6%

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

       5. +-commutative 85.6%

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

       6. unsub-neg 85.6%

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

       7. *-commutative 85.6%

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

   14. Simplified 85.6%

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

   if 4.79999999999999982 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. +-commutative 48.8%

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

      3. sqr-neg 48.8%

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

      4. unsub-neg 48.8%

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

      5. sqr-neg 48.8%

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

      6. fma-neg 48.9%

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

      7. distribute-lft-neg-in 48.9%

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

      8. *-commutative 48.9%

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

      9. *-commutative 48.9%

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

      10. distribute-rgt-neg-in 48.9%

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

      11. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 94.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   6. Taylor expanded in a around 0 94.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot c + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
   7. Step-by-step derivation

      1. neg-mul-1 94.1%

         \[\leadsto \frac{\color{blue}{\left(-c\right)} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b} \]

      2. +-commutative 94.1%

         \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \left(-c\right)}}{b} \]

      3. unsub-neg 94.1%

         \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c}}{b} \]

   8. Simplified 94.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{4}}, -{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ 1.0 (* 2.0 (/ a (- (sqrt (fma b b (* (* a c) -4.0))) b))))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = 1.0 / (2.0 * (a / (sqrt(fma(b, b, ((a * c) * -4.0))) - b)));
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(1.0 / Float64(2.0 * Float64(a / Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b))));
	else
		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)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(1.0 / N[(2.0 * N[(a / N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. add-cbrt-cube 84.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 84.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 84.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 81.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. fma-define 81.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      3. pow2 81.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2}}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      4. pow1/3 80.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      5. pow-pow 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      6. metadata-eval 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      7. associate-*l* 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}{a \cdot 2} \]

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 81.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}} \]

       2. inv-pow 81.2%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]

   12. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

       2. associate-/l* 85.6%

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

       3. fma-undefine 85.6%

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

       4. neg-mul-1 85.6%

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

       5. +-commutative 85.6%

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

       6. unsub-neg 85.6%

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

       7. *-commutative 85.6%

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

   14. Simplified 85.6%

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

   if 4.79999999999999982 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. +-commutative 48.8%

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

      3. sqr-neg 48.8%

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

      4. unsub-neg 48.8%

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

      5. sqr-neg 48.8%

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

      6. fma-neg 48.9%

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

      7. distribute-lft-neg-in 48.9%

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

      8. *-commutative 48.9%

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

      9. *-commutative 48.9%

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

      10. distribute-rgt-neg-in 48.9%

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

      11. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in c around 0 93.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.0)
   (/ 1.0 (* 2.0 (/ a (- (sqrt (fma b b (* (* a c) -4.0))) b))))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.0) {
		tmp = 1.0 / (2.0 * (a / (sqrt(fma(b, b, ((a * c) * -4.0))) - b)));
	} else {
		tmp = (-c - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.0)
		tmp = Float64(1.0 / Float64(2.0 * Float64(a / Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b))));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.0], N[(1.0 / N[(2.0 * N[(a / N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. add-cbrt-cube 84.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 82.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 84.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 84.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 81.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. fma-define 81.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      3. pow2 81.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2}}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      4. pow1/3 80.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      5. pow-pow 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      6. metadata-eval 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      7. associate-*l* 81.2%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}{a \cdot 2} \]

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 81.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}} \]

       2. inv-pow 81.2%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]

   12. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

       2. associate-/l* 85.6%

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

       3. fma-undefine 85.6%

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

       4. neg-mul-1 85.6%

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

       5. +-commutative 85.6%

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

       6. unsub-neg 85.6%

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

       7. *-commutative 85.6%

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

   14. Simplified 85.6%

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

   if 5 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

      2. +-commutative 48.8%

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

      3. sqr-neg 48.8%

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

      4. unsub-neg 48.8%

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

      5. sqr-neg 48.8%

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

      6. fma-neg 48.9%

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

      7. distribute-lft-neg-in 48.9%

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

      8. *-commutative 48.9%

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

      9. *-commutative 48.9%

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

      10. distribute-rgt-neg-in 48.9%

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

      11. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in a around 0 88.6%

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

      1. mul-1-neg 88.6%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 88.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 88.6%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 88.6%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 88.6%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 88.6%

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

      1. neg-mul-1 88.6%

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

      2. mul-1-neg 88.6%

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

      3. associate-*r/ 88.6%

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

      4. unsub-neg 88.6%

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

      5. unpow2 88.6%

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

      6. unpow2 88.6%

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

      7. times-frac 88.6%

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

      8. sqr-neg 88.6%

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

      9. distribute-frac-neg2 88.6%

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

      10. distribute-frac-neg2 88.6%

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

      11. unpow2 88.6%

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

      12. distribute-frac-neg2 88.6%

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

      13. distribute-neg-frac 88.6%

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

   10. Simplified 88.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. +-commutative 84.0%

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

      3. sqr-neg 84.0%

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

      4. unsub-neg 84.0%

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

      5. sqr-neg 84.0%

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

      6. fma-neg 84.0%

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

      7. distribute-lft-neg-in 84.0%

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

      8. *-commutative 84.0%

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

      9. *-commutative 84.0%

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

      10. distribute-rgt-neg-in 84.0%

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

      11. metadata-eval 84.0%

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

   3. Simplified 84.0%

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

   if 7 < b

   1. Initial program 48.3%

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

      1. *-commutative 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. unsub-neg 48.3%

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

      5. sqr-neg 48.3%

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

      6. fma-neg 48.4%

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

      7. distribute-lft-neg-in 48.4%

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

      8. *-commutative 48.4%

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

      9. *-commutative 48.4%

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

      10. distribute-rgt-neg-in 48.4%

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

      11. metadata-eval 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in a around 0 89.1%

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

      1. mul-1-neg 89.1%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.1%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 89.1%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.1%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 89.1%

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

      1. neg-mul-1 89.1%

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

      2. mul-1-neg 89.1%

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

      3. associate-*r/ 89.1%

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

      4. unsub-neg 89.1%

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

      5. unpow2 89.1%

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

      6. unpow2 89.1%

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

      7. times-frac 89.1%

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

      8. sqr-neg 89.1%

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

      9. distribute-frac-neg2 89.1%

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

      10. distribute-frac-neg2 89.1%

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

      11. unpow2 89.1%

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

      12. distribute-frac-neg2 89.1%

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

      13. distribute-neg-frac 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5)
   (* (- (sqrt (fma b b (* (* a c) -4.0))) b) (/ 0.5 a))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (-c - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. add-cbrt-cube 83.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 80.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 80.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 80.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 83.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 83.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 79.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. fma-define 79.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

      3. pow2 79.8%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2}}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      4. pow1/3 78.7%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      5. pow-pow 79.6%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      6. metadata-eval 79.6%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

      7. associate-*l* 79.6%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}{a \cdot 2} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 79.6%

          \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}{a \cdot 2}} \]

       2. fma-undefine 79.6%

          \[\leadsto 1 \cdot \frac{\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2} \cdot \sqrt[3]{-b} + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       3. unpow2 79.6%

          \[\leadsto 1 \cdot \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right)} \cdot \sqrt[3]{-b} + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2} \]

       4. add-cube-cbrt 84.0%

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

       5. neg-mul-1 84.0%

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

       6. fma-define 84.0%

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

       7. cancel-sign-sub-inv 84.0%

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

       8. unpow2 84.0%

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

       9. fma-define 84.0%

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

       10. metadata-eval 84.0%

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

       11. *-commutative 84.0%

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

       12. *-commutative 84.0%

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

   12. Applied egg-rr 84.0%

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

       1. associate-*r/ 84.0%

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

       2. *-commutative 84.0%

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

       3. associate-*r/ 84.0%

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

       4. fma-undefine 84.0%

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

       5. neg-mul-1 84.0%

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

       6. +-commutative 84.0%

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

       7. unsub-neg 84.0%

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

       8. *-commutative 84.0%

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

       9. associate-/r* 84.0%

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

       10. metadata-eval 84.0%

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

   14. Simplified 84.0%

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

   if 6.5 < b

   1. Initial program 48.3%

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

      1. *-commutative 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. unsub-neg 48.3%

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

      5. sqr-neg 48.3%

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

      6. fma-neg 48.4%

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

      7. distribute-lft-neg-in 48.4%

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

      8. *-commutative 48.4%

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

      9. *-commutative 48.4%

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

      10. distribute-rgt-neg-in 48.4%

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

      11. metadata-eval 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in a around 0 89.1%

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

      1. mul-1-neg 89.1%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.1%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 89.1%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.1%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 89.1%

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

      1. neg-mul-1 89.1%

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

      2. mul-1-neg 89.1%

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

      3. associate-*r/ 89.1%

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

      4. unsub-neg 89.1%

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

      5. unpow2 89.1%

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

      6. unpow2 89.1%

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

      7. times-frac 89.1%

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

      8. sqr-neg 89.1%

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

      9. distribute-frac-neg2 89.1%

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

      10. distribute-frac-neg2 89.1%

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

      11. unpow2 89.1%

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

      12. distribute-frac-neg2 89.1%

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

      13. distribute-neg-frac 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5)
   (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5) {
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.5d0) then
        tmp = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (2.0d0 * a)
    else
        tmp = (-c - (a * ((c / -b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5) {
		tmp = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * Math.pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5:
		tmp = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)
	else:
		tmp = (-c - (a * math.pow((c / -b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.5)
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	else
		tmp = (-c - (a * ((c / -b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5

   1. Initial program 84.0%

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

   if 6.5 < b

   1. Initial program 48.3%

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

      1. *-commutative 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. unsub-neg 48.3%

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

      5. sqr-neg 48.3%

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

      6. fma-neg 48.4%

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

      7. distribute-lft-neg-in 48.4%

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

      8. *-commutative 48.4%

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

      9. *-commutative 48.4%

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

      10. distribute-rgt-neg-in 48.4%

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

      11. metadata-eval 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in a around 0 89.1%

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

      1. mul-1-neg 89.1%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.1%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 89.1%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.1%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 89.1%

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

      1. neg-mul-1 89.1%

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

      2. mul-1-neg 89.1%

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

      3. associate-*r/ 89.1%

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

      4. unsub-neg 89.1%

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

      5. unpow2 89.1%

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

      6. unpow2 89.1%

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

      7. times-frac 89.1%

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

      8. sqr-neg 89.1%

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

      9. distribute-frac-neg2 89.1%

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

      10. distribute-frac-neg2 89.1%

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

      11. unpow2 89.1%

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

      12. distribute-frac-neg2 89.1%

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

      13. distribute-neg-frac 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b))

C

double code(double a, double b, double c) {
	return (-c - (a * pow((c / -b), 2.0))) / 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 - (a * ((c / -b) ** 2.0d0))) / b
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

   2. +-commutative 55.9%

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

   3. sqr-neg 55.9%

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

   4. unsub-neg 55.9%

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

   5. sqr-neg 55.9%

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

   6. fma-neg 55.9%

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

   7. distribute-lft-neg-in 55.9%

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

   8. *-commutative 55.9%

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

   9. *-commutative 55.9%

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

   10. distribute-rgt-neg-in 55.9%

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

   11. metadata-eval 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in a around 0 82.2%

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

   1. mul-1-neg 82.2%

      \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

   2. unsub-neg 82.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. mul-1-neg 82.2%

      \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   4. distribute-neg-frac2 82.2%

      \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   5. associate-/l* 82.2%

      \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

7. Simplified 82.2%

   \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

8. Taylor expanded in b around inf 82.2%

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

   1. neg-mul-1 82.2%

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

   2. mul-1-neg 82.2%

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

   3. associate-*r/ 82.2%

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

   4. unsub-neg 82.2%

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

   5. unpow2 82.2%

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

   6. unpow2 82.2%

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

   7. times-frac 82.2%

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

   8. sqr-neg 82.2%

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

   9. distribute-frac-neg2 82.2%

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

   10. distribute-frac-neg2 82.2%

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

   11. unpow2 82.2%

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

   12. distribute-frac-neg2 82.2%

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

   13. distribute-neg-frac 82.2%

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

10. Simplified 82.2%

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

11. Final simplification 82.2%

    \[\leadsto \frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b} \]
12. Add Preprocessing

Alternative 11: 64.2% accurate, 29.0× 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 / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

   2. +-commutative 55.9%

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

   3. sqr-neg 55.9%

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

   4. unsub-neg 55.9%

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

   5. sqr-neg 55.9%

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

   6. fma-neg 55.9%

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

   7. distribute-lft-neg-in 55.9%

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

   8. *-commutative 55.9%

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

   9. *-commutative 55.9%

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

   10. distribute-rgt-neg-in 55.9%

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

   11. metadata-eval 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 64.2%

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

   1. associate-*r/ 64.2%

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

   2. mul-1-neg 64.2%

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

7. Simplified 64.2%

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

8. Final simplification 64.2%

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

Alternative 12: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cbrt-cube 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 52.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 52.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 52.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 52.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 52.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 52.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 55.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 55.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. add-cube-cbrt 52.5%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. fma-define 52.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2} \]

   3. pow2 52.8%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{-b}\right)}^{2}}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

   4. pow1/3 52.0%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

   5. pow-pow 52.9%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

   6. metadata-eval 52.9%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2} \]

   7. associate-*l* 52.9%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}{a \cdot 2} \]

10. Applied egg-rr 52.9%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{-b}\right)}^{2}, \sqrt[3]{-b}, \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}{a \cdot 2} \]

11. Taylor expanded in b around inf 3.2%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b \cdot \left(1 + {\left(\sqrt[3]{-1}\right)}^{3}\right)}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b \cdot \left(1 + {\left(\sqrt[3]{-1}\right)}^{3}\right)\right)}{a}} \]

    2. rem-cube-cbrt 3.2%

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

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

    4. mul0-rgt 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

    6. rem-cube-cbrt 3.2%

       \[\leadsto \frac{0.5 \cdot \left(1 + \color{blue}{{\left(\sqrt[3]{-1}\right)}^{3}}\right)}{a} \]

    7. rem-cube-cbrt 3.2%

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

    8. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    9. metadata-eval 3.2%

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

    10. div0 3.2%

        \[\leadsto \color{blue}{0} \]

13. Simplified 3.2%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))