Quadratic roots, narrow range

Percentage Accurate: 56.1% → 91.7%

Time: 21.2s

Alternatives: 13

Speedup: 29.0×

• could not determine a ground truth

  1. a = 43128.39244788701
  2. b = 3357.8496809509993
  3. c = 12.610294166363344

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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \mathsf{fma}\left(-2, t\_0, b\right)\\ t_2 := \mathsf{fma}\left(2, t\_0, b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, t\_2, -{b}^{2}\right)}{b + \sqrt{t\_1 \cdot t\_2}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))) (t_1 (fma -2.0 t_0 b)) (t_2 (fma 2.0 t_0 b)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -60.0)
     (/ (/ (fma t_1 t_2 (- (pow b 2.0))) (+ b (sqrt (* t_1 t_2)))) (* a 2.0))
     (-
      (*
       (pow c 4.0)
       (-
        (* -5.0 (/ (pow a 3.0) (pow b 7.0)))
        (/ (+ (* 2.0 (/ (pow a 2.0) (pow b 5.0))) (/ a (* c (pow b 3.0)))) c)))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = fma(-2.0, t_0, b);
	double t_2 = fma(2.0, t_0, b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = (fma(t_1, t_2, -pow(b, 2.0)) / (b + sqrt((t_1 * t_2)))) / (a * 2.0);
	} else {
		tmp = (pow(c, 4.0) * ((-5.0 * (pow(a, 3.0) / pow(b, 7.0))) - (((2.0 * (pow(a, 2.0) / pow(b, 5.0))) + (a / (c * pow(b, 3.0)))) / c))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = fma(-2.0, t_0, b)
	t_2 = fma(2.0, t_0, b)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -60.0)
		tmp = Float64(Float64(fma(t_1, t_2, Float64(-(b ^ 2.0))) / Float64(b + sqrt(Float64(t_1 * t_2)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 3.0) / (b ^ 7.0))) - Float64(Float64(Float64(2.0 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(a / Float64(c * (b ^ 3.0)))) / c))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * t$95$0 + b), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * t$95$0 + b), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -60.0], N[(N[(N[(t$95$1 * t$95$2 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(b + N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -60

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 89.8%

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

      2. difference-of-squares 90.1%

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

      3. associate-*l* 90.1%

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

      4. sqrt-prod 90.1%

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

      5. metadata-eval 90.1%

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

      6. associate-*l* 90.1%

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

      7. sqrt-prod 90.1%

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

      8. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

   9. Taylor expanded in a around 0 90.1%

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

       1. flip-- 90.8%

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

   11. Applied egg-rr 91.6%

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

   if -60 < (/.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 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around -inf 94.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right), \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right), -{b}^{2}\right)}{b + \sqrt{\mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \mathsf{fma}\left(-2, t\_0, b\right) \cdot \mathsf{fma}\left(2, t\_0, b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\frac{\frac{t\_1 - {b}^{2}}{b + \sqrt{t\_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))) (t_1 (* (fma -2.0 t_0 b) (fma 2.0 t_0 b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -60.0)
     (/ (/ (- t_1 (pow b 2.0)) (+ b (sqrt t_1))) (* a 2.0))
     (-
      (*
       (pow c 4.0)
       (-
        (* -5.0 (/ (pow a 3.0) (pow b 7.0)))
        (/ (+ (* 2.0 (/ (pow a 2.0) (pow b 5.0))) (/ a (* c (pow b 3.0)))) c)))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = fma(-2.0, t_0, b) * fma(2.0, t_0, b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = ((t_1 - pow(b, 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = (pow(c, 4.0) * ((-5.0 * (pow(a, 3.0) / pow(b, 7.0))) - (((2.0 * (pow(a, 2.0) / pow(b, 5.0))) + (a / (c * pow(b, 3.0)))) / c))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = Float64(fma(-2.0, t_0, b) * fma(2.0, t_0, b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -60.0)
		tmp = Float64(Float64(Float64(t_1 - (b ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 3.0) / (b ^ 7.0))) - Float64(Float64(Float64(2.0 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(a / Float64(c * (b ^ 3.0)))) / c))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * t$95$0 + b), $MachinePrecision] * N[(2.0 * t$95$0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -60.0], N[(N[(N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -60

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 89.8%

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

      2. difference-of-squares 90.1%

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

      3. associate-*l* 90.1%

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

      4. sqrt-prod 90.1%

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

      5. metadata-eval 90.1%

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

      6. associate-*l* 90.1%

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

      7. sqrt-prod 90.1%

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

      8. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

      1. flip-+ 90.8%

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

   10. Applied egg-rr 91.0%

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

       1. unpow2 91.0%

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

       2. sqr-neg 91.0%

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

       3. unpow2 91.0%

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

       4. *-commutative 91.0%

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

       5. *-commutative 91.0%

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

       6. *-commutative 91.0%

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

       7. *-commutative 91.0%

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

       8. *-commutative 91.0%

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

       9. *-commutative 91.0%

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

   12. Simplified 91.0%

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

   if -60 < (/.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 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around -inf 94.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right) \cdot \mathsf{fma}\left(2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -60.0)
     (cbrt
      (pow
       (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
       3.0))
     (-
      (*
       (pow c 4.0)
       (-
        (* -5.0 (/ (pow a 3.0) (pow b 7.0)))
        (/ (+ (* 2.0 (/ (pow a 2.0) (pow b 5.0))) (/ a (* c (pow b 3.0)))) c)))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = cbrt(pow(((sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = (pow(c, 4.0) * ((-5.0 * (pow(a, 3.0) / pow(b, 7.0))) - (((2.0 * (pow(a, 2.0) / pow(b, 5.0))) + (a / (c * pow(b, 3.0)))) / c))) - (c / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = Math.cbrt(Math.pow(((Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = (Math.pow(c, 4.0) * ((-5.0 * (Math.pow(a, 3.0) / Math.pow(b, 7.0))) - (((2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + (a / (c * Math.pow(b, 3.0)))) / c))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -60.0)
		tmp = cbrt((Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0)) ^ 3.0));
	else
		tmp = Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 3.0) / (b ^ 7.0))) - Float64(Float64(Float64(2.0 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(a / Float64(c * (b ^ 3.0)))) / c))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -60.0], N[Power[N[Power[N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -60

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 89.8%

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

      2. difference-of-squares 90.1%

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

      3. associate-*l* 90.1%

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

      4. sqrt-prod 90.1%

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

      5. metadata-eval 90.1%

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

      6. associate-*l* 90.1%

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

      7. sqrt-prod 90.1%

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

      8. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

      1. add-cbrt-cube 90.1%

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

      2. pow3 90.3%

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

   10. Applied egg-rr 90.3%

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

   11. Taylor expanded in c around 0 90.3%

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

   if -60 < (/.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 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around -inf 94.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -60.0)
     (cbrt
      (pow
       (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
       3.0))
     (*
      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 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = cbrt(pow(((sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} 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;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -60.0) {
		tmp = Math.cbrt(Math.pow(((Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} 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 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -60.0)
		tmp = cbrt((Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0)) ^ 3.0));
	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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -60.0], N[Power[N[Power[N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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 (/.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)) < -60

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 89.8%

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

      2. difference-of-squares 90.1%

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

      3. associate-*l* 90.1%

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

      4. sqrt-prod 90.1%

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

      5. metadata-eval 90.1%

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

      6. associate-*l* 90.1%

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

      7. sqrt-prod 90.1%

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

      8. metadata-eval 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

      1. add-cbrt-cube 90.1%

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

      2. pow3 90.3%

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

   10. Applied egg-rr 90.3%

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

   11. Taylor expanded in c around 0 90.3%

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

   if -60 < (/.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 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in c around 0 93.8%

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

   7. Simplified 93.8%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

   8. Taylor expanded in c around 0 93.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -60:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{c}^{2} \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
     (cbrt
      (pow
       (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
       3.0))
     (-
      (*
       (pow c 2.0)
       (- (/ (* -2.0 (* c (pow a 2.0))) (pow b 5.0)) (/ a (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = cbrt(pow(((sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = (pow(c, 2.0) * (((-2.0 * (c * pow(a, 2.0))) / pow(b, 5.0)) - (a / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = Math.cbrt(Math.pow(((Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = (Math.pow(c, 2.0) * (((-2.0 * (c * Math.pow(a, 2.0))) / Math.pow(b, 5.0)) - (a / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = cbrt((Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0)) ^ 3.0));
	else
		tmp = Float64(Float64((c ^ 2.0) * Float64(Float64(Float64(-2.0 * Float64(c * (a ^ 2.0))) / (b ^ 5.0)) - Float64(a / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[Power[N[Power[N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(N[(-2.0 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -30

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 88.7%

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

      2. difference-of-squares 89.0%

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

      3. associate-*l* 89.0%

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

      4. sqrt-prod 89.0%

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

      5. metadata-eval 89.0%

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

      6. associate-*l* 89.0%

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

      7. sqrt-prod 89.0%

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

      8. metadata-eval 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. metadata-eval 89.0%

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

   8. Simplified 89.0%

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

      1. add-cbrt-cube 89.0%

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

      2. pow3 89.2%

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

   10. Applied egg-rr 89.2%

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

   11. Taylor expanded in c around 0 89.2%

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

   if -30 < (/.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 51.9%

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

      1. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around -inf 94.0%

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

   7. Taylor expanded in c around 0 91.4%

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

      1. mul-1-neg 91.4%

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

      2. unsub-neg 91.4%

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

      3. associate-*r/ 91.4%

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

      4. *-commutative 91.4%

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

   9. Simplified 91.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{c}^{2} \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
     (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
     (-
      (*
       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 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} 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 = sqrt((a * c))
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-30.0d0)) then
        tmp = (sqrt(((b + ((-2.0d0) * t_0)) * (b + (2.0d0 * t_0)))) - b) / (a * 2.0d0)
    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.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} 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.sqrt((a * c))
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0:
		tmp = (math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)
	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 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(Float64(-2.0 * 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 = sqrt((a * c));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0)
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	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[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 (/.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)) < -30

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 88.7%

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

      2. difference-of-squares 89.0%

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

      3. associate-*l* 89.0%

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

      4. sqrt-prod 89.0%

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

      5. metadata-eval 89.0%

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

      6. associate-*l* 89.0%

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

      7. sqrt-prod 89.0%

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

      8. metadata-eval 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. metadata-eval 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in a around 0 89.0%

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

   if -30 < (/.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 51.9%

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

      1. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in a around 0 91.4%

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

      1. mul-1-neg 91.4%

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

      2. unsub-neg 91.4%

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

      3. associate-*r/ 91.4%

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

   8. Simplified 91.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{2} \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
     (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
     (-
      (*
       (pow c 2.0)
       (- (/ (* -2.0 (* c (pow a 2.0))) (pow b 5.0)) (/ a (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = (pow(c, 2.0) * (((-2.0 * (c * pow(a, 2.0))) / pow(b, 5.0)) - (a / 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 = sqrt((a * c))
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-30.0d0)) then
        tmp = (sqrt(((b + ((-2.0d0) * t_0)) * (b + (2.0d0 * t_0)))) - b) / (a * 2.0d0)
    else
        tmp = ((c ** 2.0d0) * ((((-2.0d0) * (c * (a ** 2.0d0))) / (b ** 5.0d0)) - (a / (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.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = (Math.pow(c, 2.0) * (((-2.0 * (c * Math.pow(a, 2.0))) / Math.pow(b, 5.0)) - (a / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * c))
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0:
		tmp = (math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)
	else:
		tmp = (math.pow(c, 2.0) * (((-2.0 * (c * math.pow(a, 2.0))) / math.pow(b, 5.0)) - (a / math.pow(b, 3.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64((c ^ 2.0) * Float64(Float64(Float64(-2.0 * Float64(c * (a ^ 2.0))) / (b ^ 5.0)) - Float64(a / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * c));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0)
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	else
		tmp = ((c ^ 2.0) * (((-2.0 * (c * (a ^ 2.0))) / (b ^ 5.0)) - (a / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(N[(-2.0 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -30

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 88.7%

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

      2. difference-of-squares 89.0%

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

      3. associate-*l* 89.0%

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

      4. sqrt-prod 89.0%

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

      5. metadata-eval 89.0%

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

      6. associate-*l* 89.0%

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

      7. sqrt-prod 89.0%

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

      8. metadata-eval 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. metadata-eval 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in a around 0 89.0%

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

   if -30 < (/.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 51.9%

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

      1. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 94.0%

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

   6. Taylor expanded in c around -inf 94.0%

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

   7. Taylor expanded in c around 0 91.4%

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

      1. mul-1-neg 91.4%

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

      2. unsub-neg 91.4%

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

      3. associate-*r/ 91.4%

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

      4. *-commutative 91.4%

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

   9. Simplified 91.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{2} \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot t\_0\right) \cdot \left(b + 2 \cdot t\_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -30.0)
     (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
     (*
      c
      (+
       (* c (- (/ (* -2.0 (* c (pow a 2.0))) (pow b 5.0)) (/ a (pow b 3.0))))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * (((-2.0 * (c * pow(a, 2.0))) / pow(b, 5.0)) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

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((a * c))
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-30.0d0)) then
        tmp = (sqrt(((b + ((-2.0d0) * t_0)) * (b + (2.0d0 * t_0)))) - b) / (a * 2.0d0)
    else
        tmp = c * ((c * ((((-2.0d0) * (c * (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.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0) {
		tmp = (Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * (((-2.0 * (c * 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.sqrt((a * c))
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0:
		tmp = (math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)
	else:
		tmp = c * ((c * (((-2.0 * (c * 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 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -30.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(Float64(-2.0 * Float64(c * (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 = sqrt((a * c));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -30.0)
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	else
		tmp = c * ((c * (((-2.0 * (c * (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[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -30.0], N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(N[(-2.0 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 (/.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)) < -30

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 88.7%

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

      2. difference-of-squares 89.0%

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

      3. associate-*l* 89.0%

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

      4. sqrt-prod 89.0%

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

      5. metadata-eval 89.0%

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

      6. associate-*l* 89.0%

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

      7. sqrt-prod 89.0%

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

      8. metadata-eval 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. metadata-eval 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in a around 0 89.0%

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

   if -30 < (/.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 51.9%

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

      1. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in c around 0 93.9%

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

   7. Simplified 93.9%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

   8. Taylor expanded in c around 0 91.2%

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

      1. mul-1-neg 91.2%

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

      2. unsub-neg 91.2%

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

      3. associate-*r/ 91.2%

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

      4. *-commutative 91.2%

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

   10. Simplified 91.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -30:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.17:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.17)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (/ c (- b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.17) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} 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(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.17)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.17], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $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)) < -0.170000000000000012

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. +-commutative 82.0%

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

      3. sqr-neg 82.0%

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

      4. unsub-neg 82.0%

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

      5. sqr-neg 82.0%

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

      6. fma-neg 82.3%

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

      7. distribute-lft-neg-in 82.3%

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

      8. *-commutative 82.3%

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

      9. *-commutative 82.3%

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

      10. distribute-rgt-neg-in 82.3%

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

      11. metadata-eval 82.3%

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

   3. Simplified 82.3%

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

   if -0.170000000000000012 < (/.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 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. mul-1-neg 88.3%

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

      4. distribute-neg-frac2 88.3%

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

      5. associate-/l* 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in b around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. mul-1-neg 88.3%

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

      4. associate-/l* 88.3%

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

      5. unpow2 88.3%

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

      6. unpow2 88.3%

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

      7. times-frac 88.3%

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

      8. sqr-neg 88.3%

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

      9. distribute-frac-neg2 88.3%

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

      10. distribute-frac-neg2 88.3%

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

      11. unpow2 88.3%

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

      12. distribute-frac-neg2 88.3%

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

      13. mul-1-neg 88.3%

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

      14. associate-*r/ 88.3%

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

      15. mul-1-neg 88.3%

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

   10. Simplified 88.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.17:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.17) 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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.17) {
		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) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.17d0)) 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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.17) {
		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) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.17:
		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(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.17)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.17)
		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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.17], 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)) < -0.170000000000000012

   1. Initial program 82.0%

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

   if -0.170000000000000012 < (/.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 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in a around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. mul-1-neg 88.3%

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

      4. distribute-neg-frac2 88.3%

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

      5. associate-/l* 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in b around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. mul-1-neg 88.3%

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

      4. associate-/l* 88.3%

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

      5. unpow2 88.3%

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

      6. unpow2 88.3%

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

      7. times-frac 88.3%

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

      8. sqr-neg 88.3%

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

      9. distribute-frac-neg2 88.3%

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

      10. distribute-frac-neg2 88.3%

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

      11. unpow2 88.3%

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

      12. distribute-frac-neg2 88.3%

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

      13. mul-1-neg 88.3%

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

      14. associate-*r/ 88.3%

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

      15. mul-1-neg 88.3%

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

   10. Simplified 88.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.17:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in a around 0 82.1%

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

   1. mul-1-neg 82.1%

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

   2. unsub-neg 82.1%

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

   3. mul-1-neg 82.1%

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

   4. distribute-neg-frac2 82.1%

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

   5. associate-/l* 82.1%

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

7. Simplified 82.1%

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

8. Taylor expanded in b around inf 82.2%

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

   1. mul-1-neg 82.2%

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

   2. unsub-neg 82.2%

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

   3. mul-1-neg 82.2%

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

   4. associate-/l* 82.2%

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

   5. unpow2 82.2%

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

   6. unpow2 82.2%

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

   7. times-frac 82.2%

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

   8. sqr-neg 82.2%

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

   9. distribute-frac-neg2 82.2%

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

   10. distribute-frac-neg2 82.2%

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

   11. unpow2 82.2%

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

   12. distribute-frac-neg2 82.2%

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

   13. mul-1-neg 82.2%

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

   14. associate-*r/ 82.2%

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

   15. mul-1-neg 82.2%

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

10. Simplified 82.2%

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

11. Final simplification 82.2%

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

Alternative 12: 63.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. mul-1-neg 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 13: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. add-sqr-sqrt 55.8%

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

   2. difference-of-squares 55.9%

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

   3. associate-*l* 55.9%

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

   4. sqrt-prod 55.9%

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

   5. metadata-eval 55.9%

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

   6. associate-*l* 55.9%

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

   7. sqrt-prod 55.9%

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

   8. metadata-eval 55.9%

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

6. Applied egg-rr 55.9%

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

   1. *-commutative 55.9%

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

   2. cancel-sign-sub-inv 55.9%

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

   3. metadata-eval 55.9%

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

8. Simplified 55.9%

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

9. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

    2. distribute-rgt-out 3.2%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

    3. *-commutative 3.2%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{\color{blue}{c \cdot a}} \cdot \left(-2 + 2\right)\right)}{a} \]

    4. metadata-eval 3.2%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{c \cdot a} \cdot \color{blue}{0}\right)}{a} \]

    5. mul0-rgt 3.2%

       \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

    6. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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)))