Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.8%

Time: 14.7s

Alternatives: 9

Speedup: 29.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right)\\ t_1 := -{b}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, t\_0, t\_1\right)}{\mathsf{fma}\left(\sqrt{c}, \sqrt{t\_0}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{{c}^{2}}{t\_1}\right) - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 a (/ (pow b 2.0) c))) (t_1 (- (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.6)
     (/ (/ (fma c t_0 t_1) (fma (sqrt c) (sqrt t_0) b)) (* a 2.0))
     (/
      (fma
       -2.0
       (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
       (-
        (fma
         -0.25
         (* (/ (* (pow a 4.0) (pow c 4.0)) a) (/ 20.0 (pow b 6.0)))
         (* a (/ (pow c 2.0) t_1)))
        c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, (pow(b, 2.0) / c));
	double t_1 = -pow(b, 2.0);
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.6) {
		tmp = (fma(c, t_0, t_1) / fma(sqrt(c), sqrt(t_0), b)) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))), (fma(-0.25, (((pow(a, 4.0) * pow(c, 4.0)) / a) * (20.0 / pow(b, 6.0))), (a * (pow(c, 2.0) / t_1))) - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, a, Float64((b ^ 2.0) / c))
	t_1 = Float64(-(b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.6)
		tmp = Float64(Float64(fma(c, t_0, t_1) / fma(sqrt(c), sqrt(t_0), b)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), Float64(fma(-0.25, Float64(Float64(Float64((a ^ 4.0) * (c ^ 4.0)) / a) * Float64(20.0 / (b ^ 6.0))), Float64(a * Float64((c ^ 2.0) / t_1))) - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * a + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Power[b, 2.0], $MachinePrecision])}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.6], N[(N[(N[(c * t$95$0 + t$95$1), $MachinePrecision] / N[(N[Sqrt[c], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -0.599999999999999978

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in c around inf 83.8%

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

      1. flip-- 83.4%

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

      2. add-sqr-sqrt 84.1%

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

      3. fma-define 84.1%

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

      4. unpow2 84.1%

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

      5. sqrt-prod 84.0%

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

      6. fma-define 84.1%

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

      7. fma-define 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. fmm-def 85.1%

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

   9. Simplified 85.1%

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

   if -0.599999999999999978 < (/.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 48.2%

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

      1. *-commutative 48.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 94.7%

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

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, \left(-a\right) \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right), -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{c}, \sqrt{\mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{{c}^{2}}{-{b}^{2}}\right) - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, t\_0, -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{c}, \sqrt{t\_0}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 a (/ (pow b 2.0) c))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.6)
     (/ (/ (fma c t_0 (- (pow b 2.0))) (fma (sqrt c) (sqrt t_0) b)) (* a 2.0))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (* (/ c b) (/ c b))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, (pow(b, 2.0) / c));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.6) {
		tmp = (fma(c, t_0, -pow(b, 2.0)) / fma(sqrt(c), sqrt(t_0), b)) / (a * 2.0);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, a, Float64((b ^ 2.0) / c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.6)
		tmp = Float64(Float64(fma(c, t_0, Float64(-(b ^ 2.0))) / fma(sqrt(c), sqrt(t_0), b)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64(Float64(c / b) * Float64(c / b)))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * a + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.6], N[(N[(N[(c * t$95$0 + (-N[Power[b, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[Sqrt[c], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -0.599999999999999978

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in c around inf 83.8%

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

      1. flip-- 83.4%

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

      2. add-sqr-sqrt 84.1%

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

      3. fma-define 84.1%

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

      4. unpow2 84.1%

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

      5. sqrt-prod 84.0%

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

      6. fma-define 84.1%

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

      7. fma-define 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. fmm-def 85.1%

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

   9. Simplified 85.1%

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

   if -0.599999999999999978 < (/.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 48.2%

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

      1. *-commutative 48.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in a around 0 94.7%

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

      1. associate-*r/ 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. associate-/l* 94.7%

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

       2. neg-mul-1 94.7%

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

       3. unpow2 94.7%

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

       4. unpow2 94.7%

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

       5. times-frac 94.7%

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

       6. unpow2 94.7%

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

   12. Simplified 94.7%

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

       1. unpow2 94.7%

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

   14. Applied egg-rr 94.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right), -{b}^{2}\right)}{\mathsf{fma}\left(\sqrt{c}, \sqrt{\mathsf{fma}\left(-4, a, \frac{{b}^{2}}{c}\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := {b}^{2} - t\_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{t\_1 - {\left(-b\right)}^{2}}{b + \sqrt{t\_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (- (pow b 2.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0)) -0.6)
     (/ (/ (- t_1 (pow (- b) 2.0)) (+ b (sqrt t_1))) (* a 2.0))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (* (/ c b) (/ c b))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = pow(b, 2.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6) {
		tmp = ((t_1 - pow(-b, 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - ((c / b) * (c / b)))) - 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) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = (b ** 2.0d0) - t_0
    if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0d0)) <= (-0.6d0)) then
        tmp = ((t_1 - (-b ** 2.0d0)) / (b + sqrt(t_1))) / (a * 2.0d0)
    else
        tmp = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c / b) * (c / b)))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = Math.pow(b, 2.0) - t_0;
	double tmp;
	if (((Math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6) {
		tmp = ((t_1 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	t_1 = math.pow(b, 2.0) - t_0
	tmp = 0
	if ((math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6:
		tmp = ((t_1 - math.pow(-b, 2.0)) / (b + math.sqrt(t_1))) / (a * 2.0)
	else:
		tmp = ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = Float64((b ^ 2.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0)) <= -0.6)
		tmp = Float64(Float64(Float64(t_1 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64(Float64(c / b) * Float64(c / b)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	t_1 = (b ^ 2.0) - t_0;
	tmp = 0.0;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6)
		tmp = ((t_1 - (-b ^ 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	else
		tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c / b) * (c / b)))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.6], 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[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $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)) < -0.599999999999999978

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. add-cbrt-cube 82.9%

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

      2. pow1/3 78.6%

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

      3. pow3 78.6%

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

      4. pow2 78.6%

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

      5. pow-pow 78.7%

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

      6. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. flip-+ 78.6%

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

      2. pow2 78.6%

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

      3. add-sqr-sqrt 78.5%

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

      4. pow-pow 84.7%

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

      5. metadata-eval 84.7%

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

      6. *-commutative 84.7%

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

      7. *-commutative 84.7%

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

      8. pow-pow 85.0%

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

      9. metadata-eval 85.0%

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

      10. *-commutative 85.0%

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

      11. *-commutative 85.0%

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

   8. Applied egg-rr 85.0%

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

   if -0.599999999999999978 < (/.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 48.2%

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

      1. *-commutative 48.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in a around 0 94.7%

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

      1. associate-*r/ 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. associate-/l* 94.7%

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

       2. neg-mul-1 94.7%

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

       3. unpow2 94.7%

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

       4. unpow2 94.7%

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

       5. times-frac 94.7%

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

       6. unpow2 94.7%

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

   12. Simplified 94.7%

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

       1. unpow2 94.7%

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

   14. Applied egg-rr 94.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(4 \cdot a\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := {b}^{2} - t\_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{t\_1 - {\left(-b\right)}^{2}}{b + \sqrt{t\_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (- (pow b 2.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0)) -0.6)
     (/ (/ (- t_1 (pow (- b) 2.0)) (+ b (sqrt t_1))) (* a 2.0))
     (/
      (-
       (* a (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0))) (pow (/ c b) 2.0)))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = pow(b, 2.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6) {
		tmp = ((t_1 - pow(-b, 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - pow((c / b), 2.0))) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = (b ** 2.0d0) - t_0
    if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0d0)) <= (-0.6d0)) then
        tmp = ((t_1 - (-b ** 2.0d0)) / (b + sqrt(t_1))) / (a * 2.0d0)
    else
        tmp = ((a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 4.0d0))) - ((c / b) ** 2.0d0))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = Math.pow(b, 2.0) - t_0;
	double tmp;
	if (((Math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6) {
		tmp = ((t_1 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = ((a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) - Math.pow((c / b), 2.0))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	t_1 = math.pow(b, 2.0) - t_0
	tmp = 0
	if ((math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6:
		tmp = ((t_1 - math.pow(-b, 2.0)) / (b + math.sqrt(t_1))) / (a * 2.0)
	else:
		tmp = ((a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 4.0))) - math.pow((c / b), 2.0))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = Float64((b ^ 2.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0)) <= -0.6)
		tmp = Float64(Float64(Float64(t_1 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) - (Float64(c / b) ^ 2.0))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	t_1 = (b ^ 2.0) - t_0;
	tmp = 0.0;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 2.0)) <= -0.6)
		tmp = ((t_1 - (-b ^ 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	else
		tmp = ((a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 4.0))) - ((c / b) ^ 2.0))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.6], 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[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -0.599999999999999978

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. add-cbrt-cube 82.9%

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

      2. pow1/3 78.6%

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

      3. pow3 78.6%

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

      4. pow2 78.6%

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

      5. pow-pow 78.7%

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

      6. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. flip-+ 78.6%

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

      2. pow2 78.6%

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

      3. add-sqr-sqrt 78.5%

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

      4. pow-pow 84.7%

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

      5. metadata-eval 84.7%

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

      6. *-commutative 84.7%

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

      7. *-commutative 84.7%

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

      8. pow-pow 85.0%

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

      9. metadata-eval 85.0%

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

      10. *-commutative 85.0%

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

      11. *-commutative 85.0%

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

   8. Applied egg-rr 85.0%

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

   if -0.599999999999999978 < (/.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 48.2%

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

      1. *-commutative 48.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in a around 0 94.7%

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

      1. associate-*r/ 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. associate-/l* 94.7%

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

       2. neg-mul-1 94.7%

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

       3. unpow2 94.7%

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

       4. unpow2 94.7%

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

       5. times-frac 94.7%

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

       6. unpow2 94.7%

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

   12. Simplified 94.7%

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

   13. Taylor expanded in a around 0 92.2%

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

4. Final simplification 91.0%

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

Alternative 5: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.6)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (-
     (* a (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0))) (pow (/ c b) 2.0)))
     c)
    b)))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.6], 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[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.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)) < -0.599999999999999978

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

   3. Simplified 84.1%

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

   if -0.599999999999999978 < (/.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 48.2%

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

      1. *-commutative 48.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 94.7%

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

   7. Simplified 94.7%

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

   8. Taylor expanded in a around 0 94.7%

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

      1. associate-*r/ 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. associate-/l* 94.7%

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

       2. neg-mul-1 94.7%

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

       3. unpow2 94.7%

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

       4. unpow2 94.7%

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

       5. times-frac 94.7%

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

       6. unpow2 94.7%

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

   12. Simplified 94.7%

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

   13. Taylor expanded in a around 0 92.2%

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

4. Final simplification 90.9%

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

Alternative 6: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.072:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.072)
   (/ (- (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) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.072) {
		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(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.072)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / b) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.072], 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)) < -0.0719999999999999946

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

   3. Simplified 81.1%

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

   if -0.0719999999999999946 < (/.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 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in c around inf 45.5%

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

   6. Taylor expanded in b around inf 89.2%

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

      1. distribute-lft-out 89.2%

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

      2. *-lft-identity 89.2%

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

      3. times-frac 89.2%

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

      4. metadata-eval 89.2%

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

      5. *-commutative 89.2%

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

      6. associate-/l* 89.2%

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

      7. unpow2 89.2%

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

      8. unpow2 89.2%

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

      9. times-frac 89.2%

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

      10. unpow2 89.2%

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

   8. Simplified 89.2%

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

   9. Taylor expanded in b around inf 89.4%

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

       1. distribute-lft-out 89.4%

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

       2. associate-*r/ 89.4%

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

       3. mul-1-neg 89.4%

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

       4. distribute-neg-frac2 89.4%

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

       5. associate-/l* 89.4%

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

       6. unpow2 89.4%

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

       7. unpow2 89.4%

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

       8. times-frac 89.4%

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

       9. unpow2 89.4%

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

   11. Simplified 89.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.072:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.072:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.072) t_0 (/ (+ c (* a (pow (/ c b) 2.0))) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.072) {
		tmp = t_0;
	} else {
		tmp = (c + (a * pow((c / b), 2.0))) / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.072d0)) then
        tmp = t_0
    else
        tmp = (c + (a * ((c / b) ** 2.0d0))) / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.072) {
		tmp = t_0;
	} else {
		tmp = (c + (a * Math.pow((c / b), 2.0))) / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.072:
		tmp = t_0
	else:
		tmp = (c + (a * math.pow((c / b), 2.0))) / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.072)
		tmp = t_0;
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / b) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.072)
		tmp = t_0;
	else
		tmp = (c + (a * ((c / b) ^ 2.0))) / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.072], 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)) < -0.0719999999999999946

   1. Initial program 80.9%

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

   if -0.0719999999999999946 < (/.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 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in c around inf 45.5%

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

   6. Taylor expanded in b around inf 89.2%

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

      1. distribute-lft-out 89.2%

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

      2. *-lft-identity 89.2%

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

      3. times-frac 89.2%

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

      4. metadata-eval 89.2%

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

      5. *-commutative 89.2%

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

      6. associate-/l* 89.2%

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

      7. unpow2 89.2%

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

      8. unpow2 89.2%

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

      9. times-frac 89.2%

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

      10. unpow2 89.2%

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

   8. Simplified 89.2%

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

   9. Taylor expanded in b around inf 89.4%

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

       1. distribute-lft-out 89.4%

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

       2. associate-*r/ 89.4%

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

       3. mul-1-neg 89.4%

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

       4. distribute-neg-frac2 89.4%

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

       5. associate-/l* 89.4%

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

       6. unpow2 89.4%

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

       7. unpow2 89.4%

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

       8. times-frac 89.4%

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

       9. unpow2 89.4%

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

   11. Simplified 89.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.072:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.0× speedup

Math

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

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(c + Float64(a * (Float64(c / b) ^ 2.0))) / Float64(-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 54.1%

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

   1. *-commutative 54.1%

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

3. Simplified 54.3%

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

5. Taylor expanded in c around inf 53.9%

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

6. Taylor expanded in b around inf 82.4%

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

   1. distribute-lft-out 82.4%

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

   2. *-lft-identity 82.4%

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

   3. times-frac 82.4%

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

   4. metadata-eval 82.4%

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

   5. *-commutative 82.4%

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

   6. associate-/l* 82.4%

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

   7. unpow2 82.4%

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

   8. unpow2 82.4%

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

   9. times-frac 82.4%

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

   10. unpow2 82.4%

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

8. Simplified 82.4%

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

9. Taylor expanded in b around inf 82.5%

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

    1. distribute-lft-out 82.5%

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

    2. associate-*r/ 82.5%

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

    3. mul-1-neg 82.5%

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

    4. distribute-neg-frac2 82.5%

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

    5. associate-/l* 82.5%

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

    6. unpow2 82.5%

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

    7. unpow2 82.5%

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

    8. times-frac 82.5%

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

    9. unpow2 82.5%

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

11. Simplified 82.5%

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

Alternative 9: 64.3% 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 54.1%

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

   1. *-commutative 54.1%

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

3. Simplified 54.3%

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

5. Taylor expanded in b around inf 65.5%

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

   1. associate-*r/ 65.5%

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

   2. mul-1-neg 65.5%

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

7. Simplified 65.5%

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

8. Final simplification 65.5%

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

Reproduce

herbie shell --seed 2024163 
(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)))