Quadratic roots, narrow range

Percentage Accurate: 55.9% → 91.8%

Time: 13.7s

Alternatives: 10

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: 55.9% 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: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.046:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 0.046)
     (/ (/ (- 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 <= 0.046) {
		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 <= 0.046d0) 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 <= 0.046) {
		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 <= 0.046:
		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 <= 0.046)
		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 <= 0.046)
		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, 0.046], 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 < 0.045999999999999999

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

   3. Simplified 85.9%

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

      1. add-cbrt-cube 85.5%

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

      2. pow1/3 80.7%

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

      3. pow3 80.7%

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

      4. pow2 80.7%

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

      5. pow-pow 80.8%

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

      6. metadata-eval 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. flip-+ 80.9%

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

      2. pow2 80.9%

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

      3. pow-pow 82.6%

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

      4. metadata-eval 82.6%

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

      5. pow-pow 85.2%

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

      6. metadata-eval 85.2%

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

      7. add-sqr-sqrt 87.1%

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

      8. associate-*l* 87.1%

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

   8. Applied egg-rr 87.4%

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

   if 0.045999999999999999 < b

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in a around 0 93.5%

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

   6. Taylor expanded in c around 0 93.5%

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

      1. associate-*r/ 93.5%

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

      2. neg-mul-1 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. mul-1-neg 93.5%

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

      2. div-inv 93.5%

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

      3. pow-flip 93.5%

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

      4. metadata-eval 93.5%

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

   10. Applied egg-rr 93.5%

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

       1. distribute-rgt-neg-in 93.5%

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

   12. Simplified 93.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.046:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{2 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\sqrt{{b}^{4}} - t\_0} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 2.0 a)) -0.05)
     (/ (- (sqrt (- (sqrt (pow b 4.0)) t_0)) b) (* 2.0 a))
     (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.05) {
		tmp = (sqrt((sqrt(pow(b, 4.0)) - t_0)) - 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) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if (((sqrt(((b * b) - t_0)) - b) / (2.0d0 * a)) <= (-0.05d0)) then
        tmp = (sqrt((sqrt((b ** 4.0d0)) - t_0)) - 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 t_0 = c * (4.0 * a);
	double tmp;
	if (((Math.sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.05) {
		tmp = (Math.sqrt((Math.sqrt(Math.pow(b, 4.0)) - t_0)) - b) / (2.0 * a);
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	tmp = 0
	if ((math.sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.05:
		tmp = (math.sqrt((math.sqrt(math.pow(b, 4.0)) - t_0)) - b) / (2.0 * a)
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(2.0 * a)) <= -0.05)
		tmp = Float64(Float64(sqrt(Float64(sqrt((b ^ 4.0)) - t_0)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	tmp = 0.0;
	if (((sqrt(((b * b) - t_0)) - b) / (2.0 * a)) <= -0.05)
		tmp = (sqrt((sqrt((b ^ 4.0)) - t_0)) - b) / (2.0 * a);
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[Sqrt[N[(N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.050000000000000003

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. add-sqr-sqrt 80.9%

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

      2. sqrt-unprod 80.9%

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

      3. pow2 80.9%

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

      4. pow2 80.9%

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

      5. pow-prod-up 81.1%

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

      6. metadata-eval 81.1%

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

   6. Applied egg-rr 81.1%

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

   if -0.050000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in a around 0 92.3%

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

   6. Taylor expanded in b around inf 86.7%

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

      1. neg-mul-1 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. associate-/l* 86.7%

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

      5. unpow2 86.7%

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

      6. unpow2 86.7%

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

      7. times-frac 86.7%

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

      8. unpow1 86.7%

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

      9. pow-plus 86.7%

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

      10. metadata-eval 86.7%

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

   8. Simplified 86.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\sqrt{{b}^{4}} - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;b \leq 0.1:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 4.0 (* a c)))))
   (if (<= b 0.1)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (-
      (*
       a
       (-
        (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0)))
        (/ (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 <= 0.1) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (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 <= 0.1d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((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 <= 0.1) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - (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 <= 0.1:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = (a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) - (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 <= 0.1)
		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(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) - 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 <= 0.1)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = (a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 5.0))) - ((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, 0.1], 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[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 < 0.10000000000000001

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

      1. add-cbrt-cube 85.2%

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

      2. pow1/3 80.8%

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

      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 80.9%

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

      6. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. flip-+ 80.9%

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

      2. pow2 80.9%

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

      3. pow-pow 82.6%

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

      4. metadata-eval 82.6%

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

      5. pow-pow 85.2%

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

      6. metadata-eval 85.2%

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

      7. add-sqr-sqrt 87.0%

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

      8. associate-*l* 87.0%

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

   8. Applied egg-rr 87.2%

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

   if 0.10000000000000001 < b

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in a around 0 90.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.1:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.1:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.1)
   (/ (- (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a)))) b) (* 2.0 a))
   (-
    (*
     a
     (-
      (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.1) {
		tmp = (sqrt((a * ((c * -4.0) + (pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (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) :: tmp
    if (b <= 0.1d0) then
        tmp = (sqrt((a * ((c * (-4.0d0)) + ((b ** 2.0d0) / a)))) - b) / (2.0d0 * a)
    else
        tmp = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((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 tmp;
	if (b <= 0.1) {
		tmp = (Math.sqrt((a * ((c * -4.0) + (Math.pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = (a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.1:
		tmp = (math.sqrt((a * ((c * -4.0) + (math.pow(b, 2.0) / a)))) - b) / (2.0 * a)
	else:
		tmp = (a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.1)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.1)
		tmp = (sqrt((a * ((c * -4.0) + ((b ^ 2.0) / a)))) - b) / (2.0 * a);
	else
		tmp = (a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 5.0))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.1], N[(N[(N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 < 0.10000000000000001

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in a around inf 85.9%

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

   if 0.10000000000000001 < b

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in a around 0 90.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.1:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a)) -0.05)
   (/ (- (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 (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)) <= -0.05) {
		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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a)) <= -0.05)
		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 / b) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.05], 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.050000000000000003

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

   3. Simplified 81.0%

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

   if -0.050000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in a around 0 92.3%

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

   6. Taylor expanded in b around inf 86.7%

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

      1. neg-mul-1 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. associate-/l* 86.7%

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

      5. unpow2 86.7%

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

      6. unpow2 86.7%

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

      7. times-frac 86.7%

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

      8. unpow1 86.7%

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

      9. pow-plus 86.7%

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

      10. metadata-eval 86.7%

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

   8. Simplified 86.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.092:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.092)
   (/ (- (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a)))) b) (* 2.0 a))
   (-
    (*
     a
     (* (pow c 3.0) (+ (* -2.0 (/ a (pow b 5.0))) (/ -1.0 (* c (pow b 3.0))))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.092) {
		tmp = (sqrt((a * ((c * -4.0) + (pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (pow(c, 3.0) * ((-2.0 * (a / pow(b, 5.0))) + (-1.0 / (c * 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) :: tmp
    if (b <= 0.092d0) then
        tmp = (sqrt((a * ((c * (-4.0d0)) + ((b ** 2.0d0) / a)))) - b) / (2.0d0 * a)
    else
        tmp = (a * ((c ** 3.0d0) * (((-2.0d0) * (a / (b ** 5.0d0))) + ((-1.0d0) / (c * (b ** 3.0d0)))))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.092) {
		tmp = (Math.sqrt((a * ((c * -4.0) + (Math.pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (Math.pow(c, 3.0) * ((-2.0 * (a / Math.pow(b, 5.0))) + (-1.0 / (c * Math.pow(b, 3.0)))))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.092:
		tmp = (math.sqrt((a * ((c * -4.0) + (math.pow(b, 2.0) / a)))) - b) / (2.0 * a)
	else:
		tmp = (a * (math.pow(c, 3.0) * ((-2.0 * (a / math.pow(b, 5.0))) + (-1.0 / (c * math.pow(b, 3.0)))))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.092)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64((c ^ 3.0) * Float64(Float64(-2.0 * Float64(a / (b ^ 5.0))) + Float64(-1.0 / Float64(c * (b ^ 3.0)))))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.092)
		tmp = (sqrt((a * ((c * -4.0) + ((b ^ 2.0) / a)))) - b) / (2.0 * a);
	else
		tmp = (a * ((c ^ 3.0) * ((-2.0 * (a / (b ^ 5.0))) + (-1.0 / (c * (b ^ 3.0)))))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.092], N[(N[(N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-2.0 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.091999999999999998

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in a around inf 85.9%

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

   if 0.091999999999999998 < b

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in a around 0 90.5%

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

   6. Taylor expanded in c around inf 90.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.092:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.092:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \left(\frac{a}{{b}^{3}} - -2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.092)
   (/ (- (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a)))) b) (* 2.0 a))
   (*
    c
    (-
     (/ -1.0 b)
     (* c (- (/ a (pow b 3.0)) (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.092) {
		tmp = (sqrt((a * ((c * -4.0) + (pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 / b) - (c * ((a / pow(b, 3.0)) - (-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))))));
	}
	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 <= 0.092d0) then
        tmp = (sqrt((a * ((c * (-4.0d0)) + ((b ** 2.0d0) / a)))) - b) / (2.0d0 * a)
    else
        tmp = c * (((-1.0d0) / b) - (c * ((a / (b ** 3.0d0)) - ((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.092) {
		tmp = (Math.sqrt((a * ((c * -4.0) + (Math.pow(b, 2.0) / a)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 / b) - (c * ((a / Math.pow(b, 3.0)) - (-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.092:
		tmp = (math.sqrt((a * ((c * -4.0) + (math.pow(b, 2.0) / a)))) - b) / (2.0 * a)
	else:
		tmp = c * ((-1.0 / b) - (c * ((a / math.pow(b, 3.0)) - (-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.092)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(c * Float64(Float64(a / (b ^ 3.0)) - Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.092)
		tmp = (sqrt((a * ((c * -4.0) + ((b ^ 2.0) / a)))) - b) / (2.0 * a);
	else
		tmp = c * ((-1.0 / b) - (c * ((a / (b ^ 3.0)) - (-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.092], N[(N[(N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(c * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.091999999999999998

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in a around inf 85.9%

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

   if 0.091999999999999998 < b

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in c around 0 90.3%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.092:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \left(\frac{a}{{b}^{3}} - -2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a))))
   (if (<= t_0 -0.05) t_0 (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (2.0d0 * a)
    if (t_0 <= (-0.05d0)) then
        tmp = t_0
    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 t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$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]}, If[LessEqual[t$95$0, -0.05], t$95$0, 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.050000000000000003

   1. Initial program 80.9%

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

   if -0.050000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in a around 0 92.3%

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

   6. Taylor expanded in b around inf 86.7%

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

      1. neg-mul-1 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. associate-/l* 86.7%

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

      5. unpow2 86.7%

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

      6. unpow2 86.7%

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

      7. times-frac 86.7%

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

      8. unpow1 86.7%

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

      9. pow-plus 86.7%

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

      10. metadata-eval 86.7%

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

   8. Simplified 86.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% 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 / 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 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in a around 0 87.7%

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

6. Taylor expanded in b around inf 80.8%

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

   1. neg-mul-1 80.8%

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

   2. mul-1-neg 80.8%

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

   3. unsub-neg 80.8%

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

   4. associate-/l* 80.8%

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

   5. unpow2 80.8%

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

   6. unpow2 80.8%

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

   7. times-frac 80.8%

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

   8. unpow1 80.8%

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

   9. pow-plus 80.8%

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

   10. metadata-eval 80.8%

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

8. Simplified 80.8%

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

Alternative 10: 64.0% 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 56.7%

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

   1. *-commutative 56.7%

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

3. Simplified 56.9%

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

5. Taylor expanded in b around inf 63.5%

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

   1. associate-*r/ 63.5%

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

   2. mul-1-neg 63.5%

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

7. Simplified 63.5%

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

8. Final simplification 63.5%

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

Reproduce

herbie shell --seed 2024170 
(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)))