Quadratic roots, narrow range

Percentage Accurate: 56.1% → 91.5%

Time: 14.2s

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: 56.1% 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: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 3:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 3.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      (+
       (* -5.0 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 6.0)))
       (-
        (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
        (+ c (/ (* a (pow c 2.0)) (pow b 2.0)))))
      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 <= 3.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((-5.0 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 6.0))) + ((-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) - (c + ((a * pow(c, 2.0)) / pow(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) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (4.0d0 * (a * c))
    if (b <= 3.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = (((-5.0d0) * (((a ** 3.0d0) * (c ** 4.0d0)) / (b ** 6.0d0))) + (((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 4.0d0))) - (c + ((a * (c ** 2.0d0)) / (b ** 2.0d0))))) / 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 <= 3.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((-5.0 * ((Math.pow(a, 3.0) * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + ((-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) - (c + ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (4.0 * (a * c))
	tmp = 0
	if b <= 3.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = ((-5.0 * ((math.pow(a, 3.0) * math.pow(c, 4.0)) / math.pow(b, 6.0))) + ((-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 4.0))) - (c + ((a * math.pow(c, 2.0)) / math.pow(b, 2.0))))) / 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 <= 3.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-5.0 * Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) - Float64(c + Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / 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 <= 3.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = ((-5.0 * (((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))) + ((-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) - (c + ((a * (c ^ 2.0)) / (b ^ 2.0))))) / 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, 3.0], 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[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

      1. add-cbrt-cube 85.4%

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

      2. pow1/3 83.3%

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

      3. pow3 83.2%

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

      4. pow2 83.2%

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

      5. pow-pow 83.3%

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

      6. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. flip-+ 83.2%

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

      2. pow2 83.2%

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

      3. add-sqr-sqrt 83.3%

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

      4. pow-pow 87.9%

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

      5. metadata-eval 87.9%

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

      6. associate-*l* 87.9%

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

      7. pow-pow 88.1%

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

      8. metadata-eval 88.1%

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

      9. associate-*l* 88.1%

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

   8. 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 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in a around 0 94.4%

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

   6. Taylor expanded in c around inf 94.4%

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

   7. Taylor expanded in b around inf 94.5%

      \[\leadsto \color{blue}{\frac{-5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 3.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (-
      (*
       a
       (*
        (pow c 2.0)
        (+
         (*
          c
          (+
           (* -5.0 (/ (* c (pow a 2.0)) (pow b 7.0)))
           (* -2.0 (/ a (pow b 5.0)))))
         (/ -1.0 (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 <= 3.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * (pow(c, 2.0) * ((c * ((-5.0 * ((c * pow(a, 2.0)) / pow(b, 7.0))) + (-2.0 * (a / pow(b, 5.0))))) + (-1.0 / 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 <= 3.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = (a * ((c ** 2.0d0) * ((c * (((-5.0d0) * ((c * (a ** 2.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * (a / (b ** 5.0d0))))) + ((-1.0d0) / (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 <= 3.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * (Math.pow(c, 2.0) * ((c * ((-5.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 7.0))) + (-2.0 * (a / Math.pow(b, 5.0))))) + (-1.0 / 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 <= 3.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = (a * (math.pow(c, 2.0) * ((c * ((-5.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 7.0))) + (-2.0 * (a / math.pow(b, 5.0))))) + (-1.0 / 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 <= 3.0)
		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((c ^ 2.0) * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 7.0))) + Float64(-2.0 * Float64(a / (b ^ 5.0))))) + Float64(-1.0 / (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 <= 3.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = (a * ((c ^ 2.0) * ((c * ((-5.0 * ((c * (a ^ 2.0)) / (b ^ 7.0))) + (-2.0 * (a / (b ^ 5.0))))) + (-1.0 / (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, 3.0], 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[Power[c, 2.0], $MachinePrecision] * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

      1. add-cbrt-cube 85.4%

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

      2. pow1/3 83.3%

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

      3. pow3 83.2%

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

      4. pow2 83.2%

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

      5. pow-pow 83.3%

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

      6. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. flip-+ 83.2%

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

      2. pow2 83.2%

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

      3. add-sqr-sqrt 83.3%

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

      4. pow-pow 87.9%

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

      5. metadata-eval 87.9%

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

      6. associate-*l* 87.9%

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

      7. pow-pow 88.1%

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

      8. metadata-eval 88.1%

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

      9. associate-*l* 88.1%

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

   8. 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 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in a around 0 94.4%

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

   6. Taylor expanded in c around inf 94.4%

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

   7. Taylor expanded in c around 0 94.4%

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

4. Final simplification 93.2%

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

Alternative 3: 91.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 3.0)
     (/ (/ (- 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 <= 3.0) {
		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 <= 3.0d0) 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 <= 3.0) {
		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 <= 3.0:
		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 <= 3.0)
		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 <= 3.0)
		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, 3.0], 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 < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

      1. add-cbrt-cube 85.4%

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

      2. pow1/3 83.3%

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

      3. pow3 83.2%

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

      4. pow2 83.2%

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

      5. pow-pow 83.3%

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

      6. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. flip-+ 83.2%

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

      2. pow2 83.2%

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

      3. add-sqr-sqrt 83.3%

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

      4. pow-pow 87.9%

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

      5. metadata-eval 87.9%

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

      6. associate-*l* 87.9%

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

      7. pow-pow 88.1%

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

      8. metadata-eval 88.1%

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

      9. associate-*l* 88.1%

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

   8. 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 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in a around 0 94.4%

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

   6. Taylor expanded in c around inf 94.4%

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

   7. Taylor expanded in c around 0 94.3%

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

4. Final simplification 93.1%

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

Alternative 4: 89.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 3.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (- (/ (* -2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 4.0)) c)
       (* a (pow (/ c (- b)) 2.0)))
      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 <= 3.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((-2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 4.0)) - 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) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (4.0d0 * (a * c))
    if (b <= 3.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = (((((-2.0d0) * ((a ** 2.0d0) * (c ** 3.0d0))) / (b ** 4.0d0)) - 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 t_0 = Math.pow(b, 2.0) - (4.0 * (a * c));
	double tmp;
	if (b <= 3.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((-2.0 * (Math.pow(a, 2.0) * Math.pow(c, 3.0))) / Math.pow(b, 4.0)) - c) - (a * Math.pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (4.0 * (a * c))
	tmp = 0
	if b <= 3.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = ((((-2.0 * (math.pow(a, 2.0) * math.pow(c, 3.0))) / math.pow(b, 4.0)) - c) - (a * math.pow((c / -b), 2.0))) / 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 <= 3.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 4.0)) - c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / 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 <= 3.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = ((((-2.0 * ((a ^ 2.0) * (c ^ 3.0))) / (b ^ 4.0)) - c) - (a * ((c / -b) ^ 2.0))) / 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, 3.0], 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[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - 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 < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

      1. add-cbrt-cube 85.4%

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

      2. pow1/3 83.3%

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

      3. pow3 83.2%

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

      4. pow2 83.2%

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

      5. pow-pow 83.3%

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

      6. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. flip-+ 83.2%

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

      2. pow2 83.2%

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

      3. add-sqr-sqrt 83.3%

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

      4. pow-pow 87.9%

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

      5. metadata-eval 87.9%

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

      6. associate-*l* 87.9%

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

      7. pow-pow 88.1%

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

      8. metadata-eval 88.1%

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

      9. associate-*l* 88.1%

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

   8. 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 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in c around 0 92.2%

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

   6. Taylor expanded in b around inf 92.4%

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

   8. Simplified 92.4%

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

4. Final simplification 91.6%

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

Alternative 5: 89.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.0], 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[(N[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - 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 < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.2%

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

   if 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in c around 0 92.2%

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

   6. Taylor expanded in b around inf 92.4%

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

   8. Simplified 92.4%

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

4. Final simplification 91.2%

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

Alternative 6: 89.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.0], 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[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(N[(c * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.2%

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

   if 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in a around 0 94.4%

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

   6. Taylor expanded in c around inf 94.4%

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

   7. Taylor expanded in c around 0 92.3%

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

      1. associate-*r/ 92.3%

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

      2. *-commutative 92.3%

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

      3. *-commutative 92.3%

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

      4. associate-*l* 92.3%

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

   9. Simplified 92.3%

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

4. Final simplification 91.2%

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

Alternative 7: 89.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 3

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.2%

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

   if 3 < b

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in c around 0 92.2%

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

   6. Taylor expanded in b around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. unsub-neg 92.2%

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

      3. associate-/l* 92.2%

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

      4. unpow2 92.2%

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

      5. unpow2 92.2%

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

      6. unpow2 92.2%

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

      7. times-frac 92.2%

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

      8. sqr-neg 92.2%

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

      9. distribute-frac-neg2 92.2%

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

      10. distribute-frac-neg2 92.2%

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

      11. swap-sqr 92.2%

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

      12. unpow2 92.2%

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

      13. *-commutative 92.2%

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

   8. Simplified 92.2%

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

4. Final simplification 91.1%

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

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 17.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ (- (* a (- (pow (/ c (- b)) 2.0))) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 17.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = ((a * -pow((c / -b), 2.0)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 17.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / Float64(-b)) ^ 2.0))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 17.0], 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[(N[(a * (-N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 17

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

   3. Simplified 84.2%

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

   if 17 < b

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

      3. distribute-rgt-neg-in 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in b around inf 88.7%

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

   10. Simplified 88.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a))
   (/ (- (* a (- (pow (/ c (- b)) 2.0))) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = ((a * -pow((c / -b), 2.0)) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 16.0d0) then
        tmp = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (2.0d0 * a)
    else
        tmp = ((a * -((c / -b) ** 2.0d0)) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = ((a * -Math.pow((c / -b), 2.0)) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 16.0:
		tmp = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)
	else:
		tmp = ((a * -math.pow((c / -b), 2.0)) - c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / Float64(-b)) ^ 2.0))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 16.0)
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	else
		tmp = ((a * -((c / -b) ^ 2.0)) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 16.0], 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[(N[(a * (-N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 84.1%

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

   if 16 < b

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

      3. distribute-rgt-neg-in 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in b around inf 88.7%

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

   10. Simplified 88.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. *-commutative 56.0%

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

3. Simplified 56.2%

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

5. Taylor expanded in c around 0 80.3%

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

   1. associate-*r/ 80.3%

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

   2. neg-mul-1 80.3%

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

   3. distribute-rgt-neg-in 80.3%

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

7. Simplified 80.3%

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

8. Taylor expanded in b around inf 80.4%

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

10. Simplified 80.4%

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

11. Final simplification 80.4%

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

Alternative 11: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

   1. *-commutative 56.0%

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

3. Simplified 56.2%

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

5. Taylor expanded in c around 0 80.3%

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

   1. associate-*r/ 80.3%

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

   2. neg-mul-1 80.3%

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

   3. distribute-rgt-neg-in 80.3%

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

7. Simplified 80.3%

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

8. Taylor expanded in c around 0 80.3%

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

   1. sub-neg 80.3%

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

   2. associate-*r/ 80.3%

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

   3. neg-mul-1 80.3%

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

   4. distribute-rgt-neg-in 80.3%

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

   5. associate-*r/ 80.3%

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

   6. +-commutative 80.3%

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

   7. distribute-frac-neg 80.3%

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

   8. distribute-rgt-neg-in 80.3%

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

   9. associate-/l* 80.3%

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

   10. unsub-neg 80.3%

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

   11. distribute-neg-frac 80.3%

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

   12. metadata-eval 80.3%

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

   13. associate-/l* 80.3%

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

10. Simplified 80.3%

    \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right)} \]
11. Add Preprocessing

Alternative 12: 63.7% 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 56.0%

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

   1. *-commutative 56.0%

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

3. Simplified 56.2%

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

5. Taylor expanded in b around inf 63.9%

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

   1. associate-*r/ 63.9%

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

   2. mul-1-neg 63.9%

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

7. Simplified 63.9%

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

8. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2024182 
(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)))